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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z1</font>, <font color="Maroon" title="c4">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#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">x</font>,<font color="Maroon" title="c2">y</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c3">z1</font>,<font color="Olive" title="b3">TS</font> &amp; <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c4">z2</font>,<font color="Olive" title="b3">TS</font> holds <br/><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">let </span><font color="Maroon" title="c5">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="c5">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">x</font>,<font color="Maroon" title="c2">y</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c3">z1</font>,<font color="Olive" title="b2">TS</font> &amp; <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c4">z2</font>,<font color="Olive" title="b2">TS</font> holds <br/><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">let </span><font color="Maroon" title="c6">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="c5">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="c6">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">x</font>,<font color="Maroon" title="c2">y</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c3">z1</font>,<font color="Olive" title="b1">TS</font> &amp; <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c4">z2</font>,<font color="Olive" title="b1">TS</font> holds <br/><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">let </span><font color="Maroon" title="c7">TS</font> be    <a href="rewrite3.html#L1" title="REWRITE3:struct.1">transition-system</a> of <font color="Maroon" title="c6">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="c7">TS</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> &amp; <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c7">TS</font> &amp; <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c4">z2</font>,<font color="Maroon" title="c7">TS</font> implies <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><a NAME="E1:21"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
the <a href="rewrite3.html#U1" title="REWRITE3:sel.1">Tran</a> of <font color="Maroon" title="c7">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">x</font>,<font color="Maroon" title="c2">y</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c7">TS</font> or  not <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c4">z2</font>,<font color="Maroon" title="c7">TS</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><a NAME="E2:21"/>
( <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c3">z1</font>,<font color="Maroon" title="c7">TS</font> &amp; <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="rewrite3.html#R1" title="REWRITE3:pred.1">--&gt;.</a> <font color="Maroon" title="c4">z2</font>,<font color="Maroon" title="c7">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">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z2</font></span><br/>

<a NAME="E3:21"/><span class="kw">then </span>
( <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="c2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c3">z1</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="c7">TS</font> &amp; <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="c2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c4">z2</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="c7">TS</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite3.html#D2" target="_self" title="REWRITE3:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21"/>
<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="#E1:21"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="funct_1.html#D1" title="FUNCT_1:def.1">FUNCT_1:def 1</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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