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<span class="kw">set </span><font color="Maroon" title="c4">bTS</font> =  <a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">wTS</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">tTS</font> =  the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span>;<br/>
<a NAME="E1:14_1_1"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</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 <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</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 <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span> holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y1</font>, <font color="Maroon" title="c9">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c7">x</font>,<font color="Maroon" title="c8">y1</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 <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y2</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 <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span> implies <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:14_1_1_1"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y1</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 <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:14_1_1_1"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y2</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 <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></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"> <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span>,<span class="p2">(<span class="default"><a href="flang_1.html#K9" title="FLANG_1:func.9">Lex</a> <font color="Maroon" title="c1">E</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="txt" href="#E1:14_1_1_1"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">y1</font> = <font color="Maroon" title="c8">y1</font>, <font color="Maroon" title="c12">y2</font> = <font color="Maroon" title="c9">y2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_1_1"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:14_1_1_1"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E5:14_1_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">TS</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#D1" target="_self" onmouseover="rs('fsm_3/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">xc</font>, <font color="Maroon" title="c14">xi</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:14_1_1_1"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c13">xc</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:14_1_1_1"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<font color="Maroon" title="c14">xi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="flang_1.html#K9" title="FLANG_1:func.9">Lex</a> <font color="Maroon" title="c1">E</font>
 <span class="kw">and </span><br/><a NAME="E9:14_1_1_1"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c10">x</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="c13">xc</font>,<font color="Maroon" title="c14">xi</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">xc</font> = <font color="Maroon" title="c13">xc</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1_1_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">xi</font> = <font color="Maroon" title="c14">xi</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="flang_1.html#K9" title="FLANG_1:func.9">Lex</a> <font color="Maroon" title="c1">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:14_1_1_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">xi</font> = <font color="Maroon" title="c16">xi</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">E</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">xc</font> = <font color="Maroon" title="c15">xc</font>, <font color="Maroon" title="c19">y1</font> = <font color="Maroon" title="c11">y1</font>, <font color="Maroon" title="c20">y2</font> = <font color="Maroon" title="c12">y2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">TS</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#D1" target="_self" onmouseover="rs('fsm_3/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E14:14_1_1_1"/>
( <font color="Maroon" title="c19">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">xi</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> (<font color="Maroon" title="c18">xc</font>,<font color="Maroon" title="c3">TS</font>) &amp; <font color="Maroon" title="c20">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">xi</font> <a href="rewrite3.html#K3" title="REWRITE3:func.3">-succ_of</a> (<font color="Maroon" title="c18">xc</font>,<font color="Maroon" title="c3">TS</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_1_1"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:14_1_1_1"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E9:14_1_1_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="fsm_3.html#D1" target="_self" onmouseover="rs('fsm_3/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:14_1_1_1"/>
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:14_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A6</font></span>: 
 the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span> 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="funct_1.html#D1" onmouseover="rs('funct_1/D1')" onmouseout="rh()">FUNCT_1:def 1</a></span>;<br/>
<a NAME="E3:14_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">TS</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_3.html#D1" target="_self" onmouseover="rs('fsm_3/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:14_1_1"/>
(  not  <a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  <a href="fsm_3.html#K1" title="FSM_3:func.1">_bool</a> <font color="Maroon" title="c3">TS</font> is  <a href="rewrite3.html#V2" title="REWRITE3:attr.2">deterministic</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_1"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="fsm_3.html#T11" target="_self" onmouseover="rs('fsm_3/T11')" onmouseout="rh()">Th11</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>
