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<b>let </b><font color="Maroon" title="c1">O</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font><br/> for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">G</font><br/> for <font color="Olive" title="b3">g1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">G</font> holds <br/> ( <font color="Olive" title="b3">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Olive" title="b2">A</font> iff  ex <font color="Olive" title="b4">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font> ex <font color="Olive" title="b5">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b6">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Olive" title="b2">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Olive" title="b1">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b4">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b6">C</font> &amp;  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b5">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g1</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="group_9.html#NM2" title="GROUP_9:NM.2">GroupWithOperators</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font><br/> for <font color="Olive" title="b2">g1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> holds <br/> ( <font color="Olive" title="b2">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Olive" title="b1">A</font> iff  ex <font color="Olive" title="b3">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b4">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b5">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b5">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Olive" title="b1">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b4">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b5">C</font> &amp;  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b4">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> holds <br/> ( <font color="Olive" title="b1">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> iff  ex <font color="Olive" title="b2">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b3">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b4">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">C</font> &amp;  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> ) )</span><br/><b>let </b><font color="Maroon" title="c4">g1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> iff  ex <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b2">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b3">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">C</font> &amp;  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> ) )</span><br/>







<b>thus </b><a NAME="E1:91"/>
( <font color="Maroon" title="c4">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> implies  ex <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b2">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b3">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">C</font> &amp;  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b2">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b3">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">C</font> &amp;  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> ) implies <font color="Maroon" title="c4">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:91_1"/><i><font color="Green" title="E68">A1</font></i>: 
<font color="Maroon" title="c4">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b2">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b3">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">C</font> &amp;  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b2">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b3">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b2">I</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">C</font> );<br/>
<b>reconsider </b><font color="Maroon" title="c5">B</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">b</font> where <font color="Olive" title="b1">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">b</font>] </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> <b>from </b><i><a class="ref" href="domain_1.html#S7" title="DOMAIN_1:sch.7">DOMAIN_1:sch 7</a>();<br/></i>
<a NAME="E3:91_1"/>
(  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font></span>)</span> &amp;  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="group_4.html#T11" title="GROUP_4:th.11">GROUP_4:11</a>, <a class="ref" href="group_4.html#T27" title="GROUP_4:th.27">GROUP_4:27</a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E4:91_1"/><b>then </b><i><font color="Green" title="E69">A2</font></i>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 
;<br/>
<div><i><font color="Green" title="E70">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> &amp; <font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> implies <font color="Maroon" title="c6">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:91_1_1"/><i><font color="Green" title="E71">A4</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>and </b><br/><a NAME="E2:91_1_1"/><i><font color="Green" title="E72">A5</font></i>: 
<font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><a NAME="E3:91_1_1"/><i><font color="Green" title="E73">A6</font></i>: 
 ex <font color="Olive" title="b1">d1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d1</font> &amp;  ex <font color="Olive" title="b2">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b3">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b4">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b3">I</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">C</font> ) )
 <b>by </b><i><a class="txt" href="group_9.html#E1:91_1_1"><i><font color="Green" title="E71">A4</font></i></a></i>;<br/><a NAME="E4:91_1_1"/><i><font color="Green" title="E74">A7</font></i>: 
 ex <font color="Olive" title="b1">d2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Maroon" title="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d2</font> &amp;  ex <font color="Olive" title="b2">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b3">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b4">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b3">I</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">C</font> ) )
 <b>by </b><i><a class="txt" href="group_9.html#E2:91_1_1"><i><font color="Green" title="E72">A5</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">F1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c9">I1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> , <font color="Maroon" title="c10">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E6:91_1_1"/><i><font color="Green" title="E75">A8</font></i>: 
<font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E7:91_1_1"/><i><font color="Green" title="E76">A9</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I1</font></span>)</span>
 <b>and </b><br/><a NAME="E8:91_1_1"/><i><font color="Green" title="E77">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">I1</font>
 <b>and </b><br/><a NAME="E9:91_1_1"/><i><font color="Green" title="E78">A11</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">C</font>
 <b>by </b><i><a class="txt" href="group_9.html#E3:91_1_1"><i><font color="Green" title="E73">A6</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">F2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c12">I2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> , <font color="Maroon" title="c13">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E11:91_1_1"/><i><font color="Green" title="E79">A12</font></i>: 
<font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E12:91_1_1"/><i><font color="Green" title="E80">A13</font></i>: 
<font color="Maroon" title="c7">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c12">I2</font></span>)</span>
 <b>and </b><br/><a NAME="E13:91_1_1"/><i><font color="Green" title="E81">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">I2</font>
 <b>and </b><br/><a NAME="E14:91_1_1"/><i><font color="Green" title="E82">A15</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">F2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">C</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:91_1_1"><i><font color="Green" title="E74">A7</font></i></a></i>;<br/><i><font color="Green" title="E83">A16</font></i>: <font color="Maroon" title="c6">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">d</font> = 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I1</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c12">I2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E7:91_1_1"><i><font color="Green" title="E76">A9</font></i></a>, <a class="txt" href="group_9.html#E12:91_1_1"><i><font color="Green" title="E80">A13</font></i></a>, <a class="ref" href="group_4.html#T8" title="GROUP_4:th.8">GROUP_4:8</a></i>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">F2</font></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c12">I2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E8:91_1_1"><i><font color="Green" title="E77">A10</font></i></a>, <a class="txt" href="group_9.html#E13:91_1_1"><i><font color="Green" title="E81">A14</font></i></a>, <a class="ref" href="group_4.html#T25" title="GROUP_4:th.25">GROUP_4:25</a></i>
;<br/><a NAME="E16:91_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">F2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">F1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">F2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></i>;<br/><a NAME="E17:91_1_1"/><b>then </b><i><font color="Green" title="E84">A17</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">F2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">C</font> &amp; <font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E6:91_1_1"><i><font color="Green" title="E75">A8</font></i></a>, <a class="txt" href="group_9.html#E9:91_1_1"><i><font color="Green" title="E78">A11</font></i></a>, <a class="txt" href="group_9.html#E11:91_1_1"><i><font color="Green" title="E79">A12</font></i></a>, <a class="txt" href="group_9.html#E14:91_1_1"><i><font color="Green" title="E82">A15</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c11">F2</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">I1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">I2</font></span>)</span>
<b>by </b><i><a class="txt" href="group_9.html#E8:91_1_1"><i><font color="Green" title="E77">A10</font></i></a>, <a class="txt" href="group_9.html#E13:91_1_1"><i><font color="Green" title="E81">A14</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">I1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c12">I2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
;<br/><b>hence </b><a NAME="E19:91_1_1"/>
<font color="Maroon" title="c6">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="group_9.html#E15:91_1_1"><i><font color="Green" title="E83">A16</font></i></a>, <a class="txt" href="group_9.html#E17:91_1_1"><i><font color="Green" title="E84">A17</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E71">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> implies <font color="Maroon" title="c6">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> )</span><br/><b>assume </b><a NAME="E1:91_1_2"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><a NAME="E2:91_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">d1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d1</font> &amp;  ex <font color="Olive" title="b2">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b3">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b4">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b3">I</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">C</font> ) )
 ;<br/><b>then consider </b><font color="Maroon" title="c7">F1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c8">I1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> , <font color="Maroon" title="c9">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E4:91_1_2"/><i><font color="Green" title="E72">A19</font></i>: 
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E5:91_1_2"/><i><font color="Green" title="E73">A20</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span>
 <b>and </b><br/><a NAME="E6:91_1_2"/><i><font color="Green" title="E74">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font>
 <b>and </b><br/><a NAME="E7:91_1_2"/><i><font color="Green" title="E75">A22</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 ;<br/><a NAME="E8:91_1_2"/><i><font color="Green" title="E76">A23</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">I1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:91_1_2"><i><font color="Green" title="E74">A21</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> $1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c10">F2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E10:91_1_2"/><i><font color="Green" title="E77">A24</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font>
 <b>and </b><br/><a NAME="E11:91_1_2"/><i><font color="Green" title="E78">A25</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">F2</font> holds <br/><font color="Maroon" title="c10">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i><a NAME="E12:91_1_2"/><i><font color="Green" title="E79">A27</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> $1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Olive" title="b1">i</font> );<br/><a NAME="E13:91_1_2"/><i><font color="Green" title="E80">A29</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font></span>)</span> implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<b>assume </b><a NAME="E1:91_1_2_1"/>
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<a NAME="E2:91_1_2_1"/><b>then </b><i><font color="Green" title="E81">A30</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">I1</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">n</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="group_9.html#E76"><i><font color="Green" title="E65">Lm23</font></i></a></i>;<br/>
<a NAME="E4:91_1_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font> )
 
<b>by </b><i><a class="txt" href="group_9.html#E2:91_1_2_1"><i><font color="Green" title="E81">A30</font></i></a>, <a class="txt" href="group_9.html#E76"><i><font color="Green" title="E65">Lm23</font></i></a></i>;<br/>
<a NAME="E5:91_1_2_1"/><b>then </b>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">I1</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:91_1_2_1"/><b>then </b>
( <font color="Maroon" title="c8">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">I1</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">I1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">i</font> = <font color="Maroon" title="c8">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c13">i</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c14">i</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c15">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">k</font>,<font color="Maroon" title="c15">x</font>]</span><br/>

<b>take </b>
<font color="Maroon" title="c14">i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c14">i</font> )</span><br/>

<b>thus </b><a NAME="E10:91_1_2_1"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c14">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c11">I2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E15:91_1_2"/><i><font color="Green" title="E81">A31</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font></span>)</span>
 <b>and </b><br/><a NAME="E16:91_1_2"/><i><font color="Green" title="E82">A32</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c11">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S1" title="FINSEQ_1:sch.1">FINSEQ_1:sch 1</a>(<a class="txt" href="group_9.html#E13:91_1_2"><i><font color="Green" title="E80">A29</font></i></a>);<br/></i><a NAME="E17:91_1_2"/><i><font color="Green" title="E83">A33</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">I1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E18:91_1_2"/><i><font color="Green" title="E84">A34</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">I2</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:91_1_2"><i><font color="Green" title="E74">A21</font></i></a>, <a class="txt" href="group_9.html#E10:91_1_2"><i><font color="Green" title="E77">A24</font></i></a>, <a class="txt" href="group_9.html#E15:91_1_2"><i><font color="Green" title="E81">A31</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E19:91_1_2"/><i><font color="Green" title="E85">A35</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">I2</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:91_1_2"><i><font color="Green" title="E74">A21</font></i></a>, <a class="txt" href="group_9.html#E10:91_1_2"><i><font color="Green" title="E77">A24</font></i></a>, <a class="txt" href="group_9.html#E15:91_1_2"><i><font color="Green" title="E81">A31</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E20:91_1_2"/><i><font color="Green" title="E86">A36</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F2</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F1</font> )</span><br/>

<b>assume </b><a NAME="E1:91_1_2_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F1</font></span><br/>

<b>then consider </b><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:91_1_2_2"/><i><font color="Green" title="E87">A37</font></i>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">F2</font> &amp; <font color="Maroon" title="c10">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c13">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="group_9.html#E3:91_1_2_2"><i><font color="Green" title="E87">A37</font></i></a></i>;<br/>
<a NAME="E5:91_1_2_2"/><i><font color="Green" title="E88">A38</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="group_9.html#E11:91_1_2"><i><font color="Green" title="E78">A25</font></i></a>, <a class="txt" href="group_9.html#E3:91_1_2_2"><i><font color="Green" title="E87">A37</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">n</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="group_9.html#E10:91_1_2"><i><font color="Green" title="E77">A24</font></i></a>, <a class="txt" href="group_9.html#E3:91_1_2_2"><i><font color="Green" title="E87">A37</font></i></a>, <a class="txt" href="group_9.html#E76"><i><font color="Green" title="E65">Lm23</font></i></a></i>;<br/>
<a NAME="E7:91_1_2_2"/><i><font color="Green" title="E89">A39</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E10:91_1_2"><i><font color="Green" title="E77">A24</font></i></a>, <a class="txt" href="group_9.html#E3:91_1_2_2"><i><font color="Green" title="E87">A37</font></i></a>, <a class="txt" href="group_9.html#E76"><i><font color="Green" title="E65">Lm23</font></i></a></i>;<br/>
<a NAME="E8:91_1_2_2"/>
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E10:91_1_2"><i><font color="Green" title="E77">A24</font></i></a>, <a class="txt" href="group_9.html#E3:91_1_2_2"><i><font color="Green" title="E87">A37</font></i></a>, <a class="txt" href="group_9.html#E76"><i><font color="Green" title="E65">Lm23</font></i></a></i>;<br/>
<a NAME="E9:91_1_2_2"/><b>then </b>
<font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F1</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E7:91_1_2_2"><i><font color="Green" title="E89">A39</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E10:91_1_2_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">F1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E5:91_1_2_2"><i><font color="Green" title="E88">A38</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E21:91_1_2"/><b>then </b><i><font color="Green" title="E87">A40</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">C</font>
 <b>by </b><i><a class="txt" href="group_9.html#E7:91_1_2"><i><font color="Green" title="E75">A22</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E22:91_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">F2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="group_9.html#E20:91_1_2"><i><font color="Green" title="E86">A36</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">F2</font> = <font color="Maroon" title="c10">F2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E24:91_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">I2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">I2</font> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  )</span><br/>

<b>assume </b><a NAME="E1:91_1_2_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">I2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:91_1_2_3"/><i><font color="Green" title="E88">A41</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">I2</font>
 <b>and </b><br/><a NAME="E4:91_1_2_3"/><i><font color="Green" title="E89">A42</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c14">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="group_9.html#E3:91_1_2_3"><i><font color="Green" title="E88">A41</font></i></a></i>;<br/>
<a NAME="E6:91_1_2_3"/>
 ex <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">y</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Olive" title="b1">i</font> )
 
<b>by </b><i><a class="txt" href="group_9.html#E15:91_1_2"><i><font color="Green" title="E81">A31</font></i></a>, <a class="txt" href="group_9.html#E16:91_1_2"><i><font color="Green" title="E82">A32</font></i></a>, <a class="txt" href="group_9.html#E3:91_1_2_3"><i><font color="Green" title="E88">A41</font></i></a>, <a class="txt" href="group_9.html#E4:91_1_2_3"><i><font color="Green" title="E89">A42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:91_1_2_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c13">I2</font> = <font color="Maroon" title="c11">I2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>set </b><font color="Maroon" title="c14">p</font> = <font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font>;<br/><b>set </b><font color="Maroon" title="c15">q</font> = <font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font>;<br/><a NAME="E26:91_1_2"/><i><font color="Green" title="E88">A43</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">F1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">F2</font> )
 <b>by </b><i><a class="ref" href="group_4.html#D4" title="GROUP_4:def.4">GROUP_4:def 4</a></i>;<br/><a NAME="E27:91_1_2"/><b>then </b><i><font color="Green" title="E89">A44</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">F1</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">F2</font> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> )</span><br/><b>assume </b><a NAME="E1:91_1_2_4"/><i><font color="Green" title="E90">A45</font></i>: 
<font color="Maroon" title="c16">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><br/><b>then reconsider </b><font color="Maroon" title="c17">n</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="group_9.html#E10:91_1_2"><i><font color="Green" title="E77">A24</font></i></a>, <a class="txt" href="group_9.html#E26:91_1_2"><i><font color="Green" title="E88">A43</font></i></a>, <a class="txt" href="group_9.html#E76"><i><font color="Green" title="E65">Lm23</font></i></a></i>;<br/><a NAME="E3:91_1_2_4"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">n</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c17">n</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E10:91_1_2"><i><font color="Green" title="E77">A24</font></i></a>, <a class="txt" href="group_9.html#E26:91_1_2"><i><font color="Green" title="E88">A43</font></i></a>, <a class="txt" href="group_9.html#E1:91_1_2_4"><i><font color="Green" title="E90">A45</font></i></a>, <a class="txt" href="group_9.html#E76"><i><font color="Green" title="E65">Lm23</font></i></a></i>;<br/><a NAME="E4:91_1_2_4"/><b>then </b><i><font color="Green" title="E91">A46</font></i>: 
( <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">I2</font> &amp; <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">I2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_9.html#E10:91_1_2"><i><font color="Green" title="E77">A24</font></i></a>, <a class="txt" href="group_9.html#E18:91_1_2"><i><font color="Green" title="E84">A34</font></i></a>, <a class="txt" href="group_9.html#E26:91_1_2"><i><font color="Green" title="E88">A43</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:91_1_2_4"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">I1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:91_1_2"><i><font color="Green" title="E74">A21</font></i></a>, <a class="txt" href="group_9.html#E10:91_1_2"><i><font color="Green" title="E77">A24</font></i></a>, <a class="txt" href="group_9.html#E26:91_1_2"><i><font color="Green" title="E88">A43</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E7:91_1_2_4"/><i><font color="Green" title="E92">A47</font></i>: 
( <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> &amp; <font color="Maroon" title="c13">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c18">i</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E6:91_1_2"><i><font color="Green" title="E74">A21</font></i></a>, <a class="txt" href="group_9.html#E16:91_1_2"><i><font color="Green" title="E82">A32</font></i></a>, <a class="txt" href="group_9.html#E17:91_1_2"><i><font color="Green" title="E83">A33</font></i></a>, <a class="txt" href="group_9.html#E26:91_1_2"><i><font color="Green" title="E88">A43</font></i></a>, <a class="txt" href="group_9.html#E1:91_1_2_4"><i><font color="Green" title="E90">A45</font></i></a></i>;<br/><a NAME="E8:91_1_2_4"/>
( <font color="Maroon" title="c13">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">I2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font> &amp; <font color="Maroon" title="c13">I2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">I2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_9.html#E19:91_1_2"><i><font color="Green" title="E85">A35</font></i></a>, <a class="txt" href="group_9.html#E27:91_1_2"><i><font color="Green" title="E89">A44</font></i></a>, <a class="txt" href="group_9.html#E1:91_1_2_4"><i><font color="Green" title="E90">A45</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E9:91_1_2_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c18">i</font></span>)</span> &amp; <font color="Maroon" title="c12">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> &amp; <font color="Maroon" title="c12">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> &amp; <font color="Maroon" title="c7">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E6:91_1_2"><i><font color="Green" title="E74">A21</font></i></a>, <a class="txt" href="group_9.html#E11:91_1_2"><i><font color="Green" title="E78">A25</font></i></a>, <a class="txt" href="group_9.html#E12:91_1_2"><i><font color="Green" title="E79">A27</font></i></a>, <a class="txt" href="group_9.html#E15:91_1_2"><i><font color="Green" title="E81">A31</font></i></a>, <a class="txt" href="group_9.html#E26:91_1_2"><i><font color="Green" title="E88">A43</font></i></a>, <a class="txt" href="group_9.html#E27:91_1_2"><i><font color="Green" title="E89">A44</font></i></a>, <a class="txt" href="group_9.html#E1:91_1_2_4"><i><font color="Green" title="E90">A45</font></i></a>, <a class="txt" href="group_9.html#E4:91_1_2_4"><i><font color="Green" title="E91">A46</font></i></a>, <a class="txt" href="group_9.html#E7:91_1_2_4"><i><font color="Green" title="E92">A47</font></i></a>, <a class="ref" href="group_4.html#D4" title="GROUP_4:def.4">GROUP_4:def 4</a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E10:91_1_2_4"/><b>then </b><i><font color="Green" title="E93">A48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <b>by </b><i><a class="ref" href="group_1.html#T70" title="GROUP_1:th.70">GROUP_1:70</a></i>;<br/><a NAME="E11:91_1_2_4"/>
( <font color="Maroon" title="c8">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">I1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> &amp; <font color="Maroon" title="c8">I1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">I1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_9.html#E15:91_1_2"><i><font color="Green" title="E81">A31</font></i></a>, <a class="txt" href="group_9.html#E17:91_1_2"><i><font color="Green" title="E83">A33</font></i></a>, <a class="txt" href="group_9.html#E4:91_1_2_4"><i><font color="Green" title="E91">A46</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E12:91_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c18">i</font>
 <b>by </b><i><a class="txt" href="group_9.html#E8:91_1_2"><i><font color="Green" title="E76">A23</font></i></a>, <a class="txt" href="group_9.html#E15:91_1_2"><i><font color="Green" title="E81">A31</font></i></a>, <a class="txt" href="group_9.html#E17:91_1_2"><i><font color="Green" title="E83">A33</font></i></a>, <a class="txt" href="group_9.html#E4:91_1_2_4"><i><font color="Green" title="E91">A46</font></i></a>, <a class="txt" href="group_9.html#E7:91_1_2_4"><i><font color="Green" title="E92">A47</font></i></a>, <a class="ref" href="group_4.html#D4" title="GROUP_4:def.4">GROUP_4:def 4</a></i>;<br/><a NAME="E13:91_1_2_4"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c18">i</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">k</font> )
 <b>by </b><i><a class="txt" href="group_9.html#E8:91_1_2"><i><font color="Green" title="E76">A23</font></i></a>, <a class="txt" href="group_9.html#E15:91_1_2"><i><font color="Green" title="E81">A31</font></i></a>, <a class="txt" href="group_9.html#E17:91_1_2"><i><font color="Green" title="E83">A33</font></i></a>, <a class="txt" href="group_9.html#E27:91_1_2"><i><font color="Green" title="E89">A44</font></i></a>, <a class="txt" href="group_9.html#E1:91_1_2_4"><i><font color="Green" title="E90">A45</font></i></a>, <a class="txt" href="group_9.html#E4:91_1_2_4"><i><font color="Green" title="E91">A46</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E14:91_1_2_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="group_9.html#E10:91_1_2_4"><i><font color="Green" title="E93">A48</font></i></a>, <a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">GROUP_1:19</a></i>;<br/><b>hence </b><a NAME="E15:91_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E8:91_1_2"><i><font color="Green" title="E76">A23</font></i></a>, <a class="txt" href="group_9.html#E15:91_1_2"><i><font color="Green" title="E81">A31</font></i></a>, <a class="txt" href="group_9.html#E17:91_1_2"><i><font color="Green" title="E83">A33</font></i></a>, <a class="txt" href="group_9.html#E27:91_1_2"><i><font color="Green" title="E89">A44</font></i></a>, <a class="txt" href="group_9.html#E4:91_1_2_4"><i><font color="Green" title="E91">A46</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E29:91_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c8">I1</font></span>)</span></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c13">I2</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E10:91_1_2"><i><font color="Green" title="E77">A24</font></i></a>, <a class="txt" href="group_9.html#E26:91_1_2"><i><font color="Green" title="E88">A43</font></i></a>, <a class="ref" href="group_4.html#T17" title="GROUP_4:th.17">GROUP_4:17</a></i>;<br/><b>hence </b><a NAME="E30:91_1_2"/>
<font color="Maroon" title="c6">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:91_1_2"><i><font color="Green" title="E72">A19</font></i></a>, <a class="txt" href="group_9.html#E5:91_1_2"><i><font color="Green" title="E73">A20</font></i></a>, <a class="txt" href="group_9.html#E18:91_1_2"><i><font color="Green" title="E84">A34</font></i></a>, <a class="txt" href="group_9.html#E21:91_1_2"><i><font color="Green" title="E87">A40</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><b>let </b><font color="Maroon" title="c7">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> )</span><br/><b>assume </b><a NAME="E1:91_1_3"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><a NAME="E2:91_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">d1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">d1</font> &amp;  ex <font color="Olive" title="b2">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b3">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b4">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b3">I</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">C</font> ) )
 ;<br/><b>then consider </b><font color="Maroon" title="c8">F1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c9">I1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> , <font color="Maroon" title="c10">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E4:91_1_3"/><i><font color="Green" title="E72">A49</font></i>: 
<font color="Maroon" title="c10">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E5:91_1_3"/><i><font color="Green" title="E73">A50</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I1</font></span>)</span>
 <b>and </b><br/><a NAME="E6:91_1_3"/><i><font color="Green" title="E74">A51</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">I1</font>
 <b>and </b><br/><a NAME="E7:91_1_3"/><i><font color="Green" title="E75">A52</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">C</font>
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c11">F2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E9:91_1_3"/><i><font color="Green" title="E76">A53</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F1</font>
 <b>and </b><br/><a NAME="E10:91_1_3"/><i><font color="Green" title="E77">A54</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">F2</font> holds <br/><font color="Maroon" title="c11">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>)
 <b>from </b><i><a class="ref" href="finseq_1.html#S2" title="FINSEQ_1:sch.2">FINSEQ_1:sch 2</a>();<br/></i><a NAME="E11:91_1_3"/><i><font color="Green" title="E78">A55</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">F1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E9:91_1_3"><i><font color="Green" title="E76">A53</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E12:91_1_3"/><i><font color="Green" title="E79">A56</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">F1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">F2</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font> )</span><br/><b>assume </b><a NAME="E1:91_1_3_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font></span><br/><b>then consider </b><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:91_1_3_1"/><i><font color="Green" title="E80">A57</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">F2</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:91_1_3_1"/><i><font color="Green" title="E81">A58</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">F1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E9:91_1_3"><i><font color="Green" title="E76">A53</font></i></a>, <a class="txt" href="group_9.html#E3:91_1_3_1"><i><font color="Green" title="E80">A57</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c13">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="group_9.html#E3:91_1_3_1"><i><font color="Green" title="E80">A57</font></i></a></i>;<br/><a NAME="E6:91_1_3_1"/><i><font color="Green" title="E82">A59</font></i>: 
<font color="Maroon" title="c11">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E10:91_1_3"><i><font color="Green" title="E77">A54</font></i></a>, <a class="txt" href="group_9.html#E3:91_1_3_1"><i><font color="Green" title="E80">A57</font></i></a></i>;<br/><a NAME="E7:91_1_3_1"/><i><font color="Green" title="E83">A60</font></i>: 
<font color="Maroon" title="c8">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">F1</font>
 <b>by </b><i><a class="txt" href="group_9.html#E12:91_1_3"><i><font color="Green" title="E79">A56</font></i></a>, <a class="txt" href="group_9.html#E4:91_1_3_1"><i><font color="Green" title="E81">A58</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E8:91_1_3_1"/><i><font color="Green" title="E84">A61</font></i>: 
<font color="Maroon" title="c10">C</font> <a href="group_9.html#R1" title="GROUP_9:pred.1">is_stable_under_the_action_of</a> the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:91_1_3"><i><font color="Green" title="E72">A49</font></i></a>, <a class="ref" href="group_9.html#D2" target="_self" title="GROUP_9:def.2">Def2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_3_1_1"/>
( <font color="Maroon" title="c1">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:91_1_3_1_1_1"/><i><font color="Green" title="E85">A62</font></i>: 
<font color="Maroon" title="c1">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font></span><br/><div class="add"><a NAME="E2:91_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E86">A63</font></i>: 
<font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">o</font>
 <b>by </b><i><a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></i>;<br/><b>set </b><font color="Maroon" title="c15">f</font> = the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">o</font>;<br/><b>reconsider </b><font color="Maroon" title="c16">f</font> = the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">o</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="group_9.html#E2:91_1_3_1_1_1"><i><font color="Green" title="E86">A63</font></i></a></i>;<br/><a NAME="E4:91_1_3_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:91_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E87">A64</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">C</font>
 <b>by </b><i><a class="txt" href="group_9.html#E7:91_1_3"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="group_9.html#E3:91_1_3_1"><i><font color="Green" title="E80">A57</font></i></a>, <a class="txt" href="group_9.html#E6:91_1_3_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="txt" href="group_9.html#E7:91_1_3_1"><i><font color="Green" title="E83">A60</font></i></a>, <a class="txt" href="group_9.html#E2:91_1_3_1_1_1"><i><font color="Green" title="E86">A63</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E6:91_1_3_1_1_1"/>
<font color="Maroon" title="c16">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">C</font>
 <b>by </b><i><a class="txt" href="group_9.html#E8:91_1_3_1"><i><font color="Green" title="E84">A61</font></i></a>, <a class="txt" href="group_9.html#E1:91_1_3_1_1_1"><i><font color="Green" title="E85">A62</font></i></a>, <a class="ref" href="group_9.html#D1" target="_self" title="GROUP_9:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E7:91_1_3_1_1_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font>
 <b>by </b><i><a class="txt" href="group_9.html#E5:91_1_3_1_1_1"><i><font color="Green" title="E87">A64</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:91_1_3_1_1_2"/>
<font color="Maroon" title="c1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font></span><br/><div class="add"><a NAME="E2:91_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c6">o</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></i>;<br/><a NAME="E3:91_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="group_9.html#E7:91_1_3_1"><i><font color="Green" title="E83">A60</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E4:91_1_3_1_1_2"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">C</font>
 <b>by </b><i><a class="txt" href="group_9.html#E7:91_1_3"><i><font color="Green" title="E75">A52</font></i></a>, <a class="txt" href="group_9.html#E3:91_1_3_1"><i><font color="Green" title="E80">A57</font></i></a>, <a class="txt" href="group_9.html#E6:91_1_3_1"><i><font color="Green" title="E82">A59</font></i></a>, <a class="txt" href="group_9.html#E7:91_1_3_1"><i><font color="Green" title="E83">A60</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E14:91_1_3"/><b>then </b><i><font color="Green" title="E80">A65</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">F2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">C</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E15:91_1_3"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">F2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">F2</font> = <font color="Maroon" title="c11">F2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E17:91_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c9">I1</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_9.html#E5:91_1_3"><i><font color="Green" title="E73">A50</font></i></a>, <a class="txt" href="group_9.html#E6:91_1_3"><i><font color="Green" title="E74">A51</font></i></a>, <a class="txt" href="group_9.html#E9:91_1_3"><i><font color="Green" title="E76">A53</font></i></a>, <a class="txt" href="group_9.html#E10:91_1_3"><i><font color="Green" title="E77">A54</font></i></a>, <a class="txt" href="group_9.html#E77"><i><font color="Green" title="E66">Lm24</font></i></a>, <a class="txt" href="group_9.html#E11:91_1_3"><i><font color="Green" title="E78">A55</font></i></a></i>;<br/><b>hence </b><a NAME="E18:91_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c6">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:91_1_3"><i><font color="Green" title="E72">A49</font></i></a>, <a class="txt" href="group_9.html#E6:91_1_3"><i><font color="Green" title="E74">A51</font></i></a>, <a class="txt" href="group_9.html#E9:91_1_3"><i><font color="Green" title="E76">A53</font></i></a>, <a class="txt" href="group_9.html#E14:91_1_3"><i><font color="Green" title="E80">A65</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c6">H</font> being  <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font><b> such that </b><br/><a NAME="E9:91_1"/><i><font color="Green" title="E72">A66</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:91_1"><i><font color="Green" title="E69">A2</font></i></a>, <a class="txt" href="group_9.html#E5:91_1"><i><font color="Green" title="E70">A3</font></i></a>, <a class="txt" href="group_9.html#E6:91_1"><i><font color="Green" title="E71">A18</font></i></a>, <a class="txt" href="group_9.html#E38"><i><font color="Green" title="E36">Lm15</font></i></a></i>;<br/>
<a NAME="E10:91_1"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font> )</span><br/>

<b>assume </b><a NAME="E1:91_1_4"/><i><font color="Green" title="E73">A67</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">p</font> = 1 as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> ;<br/>
<a NAME="E4:91_1_4"/><i><font color="Green" title="E74">A68</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <font color="Maroon" title="c9">p</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<i><font color="Green" title="E75">A69</font></i>:  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <font color="Maroon" title="c9">p</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span></span>)</span> = 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> 1</span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>
<b>by </b><i><a class="ref" href="group_4.html#T28" title="GROUP_4:th.28">GROUP_4:28</a></i>
<br/>.= 
<font color="Maroon" title="c8">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> 1
<b>by </b><i><a class="ref" href="group_4.html#T12" title="GROUP_4:th.12">GROUP_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c8">a</font>
<b>by </b><i><a class="ref" href="group_1.html#T44" title="GROUP_1:th.44">GROUP_1:44</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c10">C</font> =  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>;<br/>
<a NAME="E6:91_1_4"/>
( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> )
 
<b>by </b><i><a class="txt" href="group_9.html#E1:91_1_4"><i><font color="Green" title="E73">A67</font></i></a>, <a class="ref" href="group_9.html#D2" target="_self" title="GROUP_9:def.2">Def2</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E7:91_1_4"/><b>then </b>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E8:91_1_4"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 <b>by </b><i><a class="txt" href="group_9.html#E4:91_1_4"><i><font color="Green" title="E74">A68</font></i></a>, <a class="txt" href="group_9.html#E5:91_1_4"><i><font color="Green" title="E75">A69</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:91_1"/><b>then </b>
 <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c6">H</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E9:91_1"><i><font color="Green" title="E72">A66</font></i></a>, <a class="ref" href="group_9.html#D29" target="_self" title="GROUP_9:def.29">Def29</a></i>;<br/>
<a NAME="E12:91_1"/><b>then </b>
 <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c6">H</font>
 
<b>by </b><i><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></i>;<br/>
<a NAME="E13:91_1"/><b>then </b>
<font color="Maroon" title="c4">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">H</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E1:91_1"><i><font color="Green" title="E68">A1</font></i></a>, <a class="ref" href="group_2.html#T49" title="GROUP_2:th.49">GROUP_2:49</a></i>;<br/>
<a NAME="E14:91_1"/><b>then </b>
<font color="Maroon" title="c4">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E9:91_1"><i><font color="Green" title="E72">A66</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E15:91_1"/><b>then </b>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> &amp;  ex <font color="Olive" title="b2">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b3">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b4">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b3">I</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">C</font> ) )
 
;<br/>
<b>hence </b><a NAME="E16:91_1"/>
 ex <font color="Olive" title="b1">F</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> ex <font color="Olive" title="b2">I</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  ex <font color="Olive" title="b3">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">I</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">C</font> &amp;  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Olive" title="b2">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon" title="c5">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c6">I</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> , <font color="Maroon" title="c7">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font><b> such that </b><a NAME="E3:91"/><i><font color="Green" title="E68">A70</font></i>: 
<font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_9.html#K1" title="GROUP_9:func.1">the_stable_subset_generated_by</a> <font color="Maroon" title="c3">A</font>,the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E4:91"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">I</font>
 <b>and </b><br/><a NAME="E5:91"/><i><font color="Green" title="E69">A71</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C</font>
 <b>and </b><br/><a NAME="E6:91"/><i><font color="Green" title="E70">A72</font></i>: 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c6">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font></span><br/>

<b>set </b><font color="Maroon" title="c8">H'</font> =  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font>;<br/>
<a NAME="E7:91"/><i><font color="Green" title="E71">A73</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font></span>)</span> &amp; ( for <font color="Olive" title="b1">H</font> being <a href="group_9.html#V2" title="GROUP_9:attr.2">strict</a>  <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Maroon" title="c2">G</font>  st <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H</font> holds <br/> <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> is    <a href="group_9.html#M1" title="GROUP_9:mode.1">StableSubgroup</a> of <font color="Olive" title="b1">H</font> ) )
 
<b>by </b><i><a class="ref" href="group_9.html#D29" target="_self" title="GROUP_9:def.29">Def29</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">Y</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font></span>)</span>;<br/>
<a NAME="E8:91"/>
 <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">Y</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c2">G</font>  st <font color="Maroon" title="c11">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font> &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font> holds <br/><font color="Olive" title="b1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font></span><br/><b>let </b><font color="Maroon" title="c12">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c2">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font> &amp; <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font> implies <font color="Maroon" title="c12">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font> )</span><br/><b>assume </b><a NAME="E1:91_2"/><i><font color="Green" title="E72">A74</font></i>: 
<font color="Maroon" title="c11">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">O</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font> implies <font color="Maroon" title="c12">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font> )</span><br/><b>assume </b><a NAME="E2:91_2"/><i><font color="Green" title="E73">A75</font></i>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y</font> implies <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> )</span><br/><b>assume </b><a NAME="E1:91_2_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font></span><br/><b>then consider </b><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:91_2_1"/><i><font color="Green" title="E74">A76</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c14">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="group_9.html#E3:91_2_1"><i><font color="Green" title="E74">A76</font></i></a></i>;<br/><a NAME="E5:91_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="group_9.html#E3:91_2_1"><i><font color="Green" title="E74">A76</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E6:91_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="group_9.html#K3" title="GROUP_9:func.3">^</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="group_9.html#E25"><i><font color="Green" title="E24">Lm10</font></i></a></i>;<br/><a NAME="E7:91_2_1"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="group_9.html#E1:91_2"><i><font color="Green" title="E72">A74</font></i></a>, <a class="txt" href="group_9.html#E2:91_2"><i><font color="Green" title="E73">A75</font></i></a>, <a class="ref" href="group_9.html#D6" target="_self" title="GROUP_9:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E8:91_2_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Y</font>
 <b>by </b><i><a class="txt" href="group_9.html#E3:91_2_1"><i><font color="Green" title="E74">A76</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:91_2"/>
<font color="Maroon" title="c12">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E11:91"/><b>then </b><i><font color="Green" title="E72">A77</font></i>: 
<font color="Maroon" title="c10">Y</font> <a href="group_9.html#R1" title="GROUP_9:pred.1">is_stable_under_the_action_of</a> the <a href="group_9.html#U1" title="GROUP_9:sel.1">action</a> of <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="group_9.html#D1" target="_self" title="GROUP_9:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">H'</font> =  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font> as    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="group_9.html#D7" target="_self" title="GROUP_9:def.7">Def7</a></i>;<br/>
<a NAME="E13:91"/>
<font color="Maroon" title="c7">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">H'</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E3:91"><i><font color="Green" title="E68">A70</font></i></a>, <a class="txt" href="group_9.html#E7:91"><i><font color="Green" title="E71">A73</font></i></a>, <a class="txt" href="group_9.html#E11:91"><i><font color="Green" title="E72">A77</font></i></a>, <a class="ref" href="group_9.html#D2" target="_self" title="GROUP_9:def.2">Def2</a></i>;<br/>
<a NAME="E14:91"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c11">H'</font>
 
<b>by </b><i><a class="txt" href="group_9.html#E5:91"><i><font color="Green" title="E69">A71</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E15:91"/>
<font color="Maroon" title="c4">g1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_9.html#K18" title="GROUP_9:func.18">the_stable_subgroup_of</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="group_9.html#E6:91"><i><font color="Green" title="E70">A72</font></i></a>, <a class="ref" href="group_4.html#T26" title="GROUP_4:th.26">GROUP_4:26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
