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<b>let </b><font color="Maroon" title="c1">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c1">f</font> <a href="calcul_1.html#R1" title="CALCUL_1:pred.1">is_tail_of</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c1">f</font> implies  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c1">f</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c1">f</font> )</span><br/>

<b>assume </b><a NAME="E1:31"/>
 <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c1">f</font> <a href="calcul_1.html#R1" title="CALCUL_1:pred.1">is_tail_of</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c1">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c1">f</font> <a href="calcul_1.html#R10" title="CALCUL_1:pred.10">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c1">f</font></span><br/>

<b>then consider </b><font color="Maroon" title="c2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:31"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c2">i</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> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c1">f</font> )
 <b>by </b><i><a class="ref" href="calcul_1.html#D3" target="_self" title="CALCUL_1:def.3">Def3</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D15" target="_self" title="CALCUL_1:def.15">CALCUL_1:def 15</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">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c3">A</font><br/> for <font color="Olive" title="b2">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c3">A</font>  st <font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c1">f</font></span><br/><b>let </b><font color="Maroon" title="c4">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <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"> for <font color="Olive" title="b1">v</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c3">A</font>  st <font color="Maroon" title="c4">J</font>,<font color="Olive" title="b1">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c1">f</font> holds <br/><font color="Maroon" title="c4">J</font>,<font color="Olive" title="b1">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c1">f</font></span><br/><b>let </b><font color="Maroon" title="c5">v</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_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="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a>  <a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c1">f</font> implies <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c1">f</font> )</span><br/>





<b>assume </b><a NAME="E4:31"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c1">f</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="calcul_1.html#D14" target="_self" title="CALCUL_1:def.14">CALCUL_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c1">f</font></span><br/>

<a NAME="E5:31"/>
 <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c1">f</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> <span class="p1">(<span class="default"><a href="calcul_1.html#K1" title="CALCUL_1:func.1">Ant</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E3:31"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E6:31"/>
<font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a>  <a href="calcul_1.html#K2" title="CALCUL_1:func.2">Suc</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:31"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="calcul_1.html#D11" target="_self" title="CALCUL_1:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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