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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">z1</font>, <font color="Maroon" title="c5">y2</font>, <font color="Maroon" title="c6">z2</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#V2" title="REWRITE3:attr.2">deterministic</a>   <a href="rewrite3.html#L1" title="REWRITE3:struct.1">transition-system</a> of <font color="Olive" title="b2">F</font>  st <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">z1</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="c6">z2</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> &amp; <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">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="c7">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#V2" title="REWRITE3:attr.2">deterministic</a>   <a href="rewrite3.html#L1" title="REWRITE3:struct.1">transition-system</a> of <font color="Olive" title="b1">F</font>  st <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">z1</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="c6">z2</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> &amp; <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">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="c7">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#V2" title="REWRITE3:attr.2">deterministic</a>   <a href="rewrite3.html#L1" title="REWRITE3:struct.1">transition-system</a> of <font color="Maroon" title="c8">F</font>  st <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">z1</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="c6">z2</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> &amp; <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">TS</font> be   <a href="rewrite3.html#V2" title="REWRITE3:attr.2">deterministic</a>   <a href="rewrite3.html#L1" title="REWRITE3:struct.1">transition-system</a> of <font color="Maroon" title="c8">F</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="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">z1</font>,<font color="Maroon" title="c9">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="c6">z2</font>,<font color="Maroon" title="c9">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> &amp; <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font> ) )</span><br/>

















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:35"/><span class="lab"><font color="Green" title="E30">A1</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">z1</font>,<font color="Maroon" title="c9">TS</font>
 <span class="kw">and </span><br/><a NAME="E2:35"/><span class="lab"><font color="Green" title="E31">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="c5">y2</font>,<font color="Maroon" title="c6">z2</font>,<font color="Maroon" title="c9">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> &amp; <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c10">v2</font>, <font color="Maroon" title="c11">w2</font> being    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c7">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E4:35"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c10">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font>
 <span class="kw">and </span><br/><a NAME="E5:35"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c11">w2</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c9">TS</font>
 <span class="kw">and </span><br/><a NAME="E6:35"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">w2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c10">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E31">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="c12">v1</font>, <font color="Maroon" title="c13">w1</font> being    <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c7">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a> <span class="kw"> such that </span><br/><a NAME="E8:35"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c12">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z1</font>
 <span class="kw">and </span><br/><a NAME="E9:35"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c13">w1</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c3">y1</font>,<font color="Maroon" title="c9">TS</font>
 <span class="kw">and </span><br/><a NAME="E10:35"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">w1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c12">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="rewrite3.html#D3" target="_self" title="REWRITE3:def.3">Def3</a></span>;<br/>
<a NAME="E11:35"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Maroon" title="c9">TS</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite3.html#D1" target="_self" title="REWRITE3:def.1">Def1</a></span>;<br/>
<a NAME="E12:35"/>
(  ex <font color="Olive" title="b1">u9</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c7">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  st <br/>( <font color="Maroon" title="c13">w1</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b1">u9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">w2</font> &amp; <font color="Maroon" title="c12">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u9</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c10">v2</font> ) or  ex <font color="Olive" title="b1">u9</font> being   <a href="catalan2.html#M2" title="CATALAN2:mode.2">Element</a> of <font color="Maroon" title="c7">E</font> <a href="catalan2.html#K2" title="CATALAN2:func.2">^omega</a>  st <br/>( <font color="Maroon" title="c11">w2</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Olive" title="b1">u9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">w1</font> &amp; <font color="Maroon" title="c10">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u9</font> <a href="flang_1.html#K1" title="FLANG_1:func.1">^</a> <font color="Maroon" title="c12">v1</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="rewrite3.html#T13" target="_self" title="REWRITE3:th.13">Th13</a></span>;<br/>
<a NAME="E13:35"/><span class="kw">then </span>
<font color="Maroon" title="c13">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:35"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="rewrite3.html#T19" target="_self" title="REWRITE3:th.19">Th19</a></span>;<br/>
<a NAME="E14:35"/><span class="kw">then </span>
<font color="Maroon" title="c12">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:35"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E34">A5</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="E15:35"/>
( <font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y2</font> &amp; <font color="Maroon" title="c4">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E8:35"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E11:35"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="rewrite3.html#T27" target="_self" title="REWRITE3:th.27">Th27</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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