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<span class="kw">let </span><font color="Maroon" title="c1">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c1">A</font><br/>  for <font color="Olive" title="b2">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <br/>  for <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">w</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b3">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" title="CALCUL_1:func.3">still_not-bound_in</a> <font color="Olive" title="b2">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" title="CALCUL_1:func.3">still_not-bound_in</a> <font color="Olive" title="b2">f</font></span>)</span> &amp; <font color="Olive" title="b1">J</font>,<font color="Olive" title="b3">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Olive" title="b2">f</font> holds <br/><font color="Olive" title="b1">J</font>,<font color="Olive" title="b4">w</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Olive" title="b2">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <br/>  for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b2">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" title="CALCUL_1:func.3">still_not-bound_in</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" title="CALCUL_1:func.3">still_not-bound_in</a> <font color="Olive" title="b1">f</font></span>)</span> &amp; <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b2">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon" title="c2">J</font>,<font color="Olive" title="b3">w</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" title="CALCUL_1:func.3">still_not-bound_in</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" title="CALCUL_1:func.3">still_not-bound_in</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c2">J</font>,<font color="Olive" title="b1">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Maroon" title="c3">f</font> holds <br/><font color="Maroon" title="c2">J</font>,<font color="Olive" title="b2">w</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Maroon" title="c3">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="valuat_1.html#K2" title="VALUAT_1:func.2">Valuations_in</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" title="CALCUL_1:func.3">still_not-bound_in</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" title="CALCUL_1:func.3">still_not-bound_in</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c4">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">w</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Maroon" title="c3">f</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:44"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" title="CALCUL_1:func.3">still_not-bound_in</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="calcul_1.html#K3" title="CALCUL_1:func.3">still_not-bound_in</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c4">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Maroon" title="c3">f</font> or <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">w</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Maroon" title="c3">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:44"/>
<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c4">v</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">w</font> <a href="calcul_1.html#R9" title="CALCUL_1:pred.9">|=</a> <font color="Maroon" title="c3">f</font></span><br/>

<a NAME="E3:44"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A2</font></span>: 
<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c4">v</font> <a href="calcul_1.html#R6" title="CALCUL_1:pred.6">|=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="calcul_1.html#D14" target="_self" title="CALCUL_1:def.14">Def14</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="calcul_1.html#D11" target="_self" title="CALCUL_1:def.11">CALCUL_1:def 11</a>,<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font> implies <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">w</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c6">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:44"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">w</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c6">p</font></span><br/>

<a NAME="E5:44"/>
 ex <font color="Olive" title="b1">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:44"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<a NAME="E6:44"/><span class="kw">then </span>
 for <font color="Olive" title="b1">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c6">p</font> holds <br/><font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="calcul_1.html#K3" title="CALCUL_1:func.3">still_not-bound_in</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="calcul_1.html#D5" target="_self" title="CALCUL_1:def.5">Def5</a></span>;<br/>
<a NAME="E7:44"/><span class="kw">then </span>
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c6">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="calcul_1.html#K3" title="CALCUL_1:func.3">still_not-bound_in</a> <font color="Maroon" title="c3">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E8:44"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">w</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c6">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></span>;<br/>
<a NAME="E9:44"/>
<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c4">v</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c6">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E4:44"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="calcul_1.html#D11" target="_self" title="CALCUL_1:def.11">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:44"/>
<font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c5">w</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:44"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="sublemma.html#T68" title="SUBLEMMA:th.68">SUBLEMMA:68</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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