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<b>let </b><font color="Maroon" title="c1">k</font> be    <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> ; <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 non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b3">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b4">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b5">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="Olive" title="b1">A</font><br/> for <font color="Olive" title="b6">vS</font>, <font color="Olive" title="b7">vS1</font>, <font color="Olive" title="b8">vS2</font> being  <a href="sublemma.html#NM1" title="SUBLEMMA:NM.1">Val_Sub</a> of <font color="Olive" title="b1">A</font>  st ( for <font color="Olive" title="b9">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b9">y</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="Olive" title="b7">vS1</font> holds <br/> not <font color="Olive" title="b9">y</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> <span class="p1">(<span class="default"><font color="Olive" title="b3">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b4">ll</font></span>)</span> ) &amp; ( for <font color="Olive" title="b9">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b9">y</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="Olive" title="b8">vS2</font> holds <br/><font color="Olive" title="b8">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b9">y</font> ) &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b6">vS</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b8">vS2</font> holds <br/>( <font color="Olive" title="b2">J</font>,<font color="Olive" title="b5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Olive" title="b6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b3">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b4">ll</font> iff <font color="Olive" title="b2">J</font>,<font color="Olive" title="b5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b3">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b4">ll</font> )</span><br/><b>let </b><font color="Maroon" title="c2">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:()"><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="c2">A</font><br/> for <font color="Olive" title="b2">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b3">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b4">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="c2">A</font><br/> for <font color="Olive" title="b5">vS</font>, <font color="Olive" title="b6">vS1</font>, <font color="Olive" title="b7">vS2</font> being  <a href="sublemma.html#NM1" title="SUBLEMMA:NM.1">Val_Sub</a> of <font color="Maroon" title="c2">A</font>  st ( for <font color="Olive" title="b8">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b8">y</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="Olive" title="b6">vS1</font> holds <br/> not <font color="Olive" title="b8">y</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> <span class="p1">(<span class="default"><font color="Olive" title="b2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b3">ll</font></span>)</span> ) &amp; ( for <font color="Olive" title="b8">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b8">y</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="Olive" title="b7">vS2</font> holds <br/><font color="Olive" title="b7">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b8">y</font> ) &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b5">vS</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b7">vS2</font> holds <br/>( <font color="Olive" title="b1">J</font>,<font color="Olive" title="b4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Olive" title="b5">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b3">ll</font> iff <font color="Olive" title="b1">J</font>,<font color="Olive" title="b4">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b6">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b7">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b2">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b3">ll</font> )</span><br/><b>let </b><font color="Maroon" title="c3">J</font> be    <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Maroon" title="c2">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">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b2">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b3">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="c2">A</font><br/> for <font color="Olive" title="b4">vS</font>, <font color="Olive" title="b5">vS1</font>, <font color="Olive" title="b6">vS2</font> being  <a href="sublemma.html#NM1" title="SUBLEMMA:NM.1">Val_Sub</a> of <font color="Maroon" title="c2">A</font>  st ( for <font color="Olive" title="b7">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b7">y</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="Olive" title="b5">vS1</font> holds <br/> not <font color="Olive" title="b7">y</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> <span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b2">ll</font></span>)</span> ) &amp; ( for <font color="Olive" title="b7">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b7">y</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="Olive" title="b6">vS2</font> holds <br/><font color="Olive" title="b6">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">y</font> ) &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b4">vS</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b6">vS2</font> holds <br/>( <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Olive" title="b4">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b2">ll</font> iff <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b3">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b5">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b6">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Olive" title="b1">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b2">ll</font> )</span><br/><b>let </b><font color="Maroon" title="c4">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</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">ll</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</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="c2">A</font><br/> for <font color="Olive" title="b3">vS</font>, <font color="Olive" title="b4">vS1</font>, <font color="Olive" title="b5">vS2</font> being  <a href="sublemma.html#NM1" title="SUBLEMMA:NM.1">Val_Sub</a> of <font color="Maroon" title="c2">A</font>  st ( for <font color="Olive" title="b6">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b6">y</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="Olive" title="b4">vS1</font> holds <br/> not <font color="Olive" title="b6">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b1">ll</font></span>)</span> ) &amp; ( for <font color="Olive" title="b6">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b6">y</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="Olive" title="b5">vS2</font> holds <br/><font color="Olive" title="b5">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b6">y</font> ) &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b3">vS</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b5">vS2</font> holds <br/>( <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b2">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Olive" title="b3">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b1">ll</font> iff <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b2">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b4">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b5">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b1">ll</font> )</span><br/><b>let </b><font color="Maroon" title="c5">ll</font> be   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</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="c2">A</font><br/> for <font color="Olive" title="b2">vS</font>, <font color="Olive" title="b3">vS1</font>, <font color="Olive" title="b4">vS2</font> being  <a href="sublemma.html#NM1" title="SUBLEMMA:NM.1">Val_Sub</a> of <font color="Maroon" title="c2">A</font>  st ( for <font color="Olive" title="b5">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b5">y</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="Olive" title="b3">vS1</font> holds <br/> not <font color="Olive" title="b5">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font></span>)</span> ) &amp; ( for <font color="Olive" title="b5">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b5">y</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="Olive" title="b4">vS2</font> holds <br/><font color="Olive" title="b4">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">y</font> ) &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">vS</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b4">vS2</font> holds <br/>( <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b1">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Olive" title="b2">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font> iff <font color="Maroon" title="c3">J</font>,<font color="Olive" title="b1">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b3">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b4">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font> )</span><br/><b>let </b><font color="Maroon" title="c6">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="c2">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">vS</font>, <font color="Olive" title="b2">vS1</font>, <font color="Olive" title="b3">vS2</font> being  <a href="sublemma.html#NM1" title="SUBLEMMA:NM.1">Val_Sub</a> of <font color="Maroon" title="c2">A</font>  st ( for <font color="Olive" title="b4">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b4">y</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="Olive" title="b2">vS1</font> holds <br/> not <font color="Olive" title="b4">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font></span>)</span> ) &amp; ( for <font color="Olive" title="b4">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b4">y</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="Olive" title="b3">vS2</font> holds <br/><font color="Olive" title="b3">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">y</font> ) &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">vS</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b3">vS2</font> holds <br/>( <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Olive" title="b1">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font> iff <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b2">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Olive" title="b3">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font> )</span><br/><b>let </b><font color="Maroon" title="c7">vS</font>, <font color="Maroon" title="c8">vS1</font>, <font color="Maroon" title="c9">vS2</font> be   <a href="sublemma.html#NM1" title="SUBLEMMA:NM.1">Val_Sub</a> of <font color="Maroon" title="c2">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">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b1">y</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="c8">vS1</font> holds <br/> not <font color="Olive" title="b1">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b1">y</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="c9">vS2</font> holds <br/><font color="Maroon" title="c9">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> ) &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">vS2</font> implies ( <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c7">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font> iff <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c9">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font> ) )</span><br/>

















<b>assume </b><a NAME="E1:94"/><i><font color="Green" title="E73">A1</font></i>: 
( ( for <font color="Olive" title="b1">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b1">y</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="c8">vS1</font> holds <br/> not <font color="Olive" title="b1">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b1">y</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="c9">vS2</font> holds <br/><font color="Maroon" title="c9">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> ) &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">vS2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c7">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font> iff <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c9">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font> )</span><br/>

<a NAME="E2:94"/><i><font color="Green" title="E74">A2</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>  st <font color="Olive" title="b1">y</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="c8">vS1</font> holds <br/> not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <font color="Maroon" title="c5">ll</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">y</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">y</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="c8">vS1</font> implies  not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <font color="Maroon" title="c5">ll</font> )</span><br/>

<b>assume </b><a NAME="E1:94_1"/>
<font color="Maroon" title="c10">y</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="c8">vS1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <font color="Maroon" title="c5">ll</font></span><br/>

<a NAME="E2:94_1"/><b>then </b>
not <font color="Maroon" title="c10">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:94"><i><font color="Green" title="E73">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:94_1"/>
not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <font color="Maroon" title="c5">ll</font>
 <b>by </b><i><a class="ref" href="qc_lang3.html#T9" title="QC_LANG3:th.9">QC_LANG3:9</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="E3:94"/><i><font color="Green" title="E75">A3</font></i>: 
( <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font></span>)</span>,<font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c7">vS</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  iff <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c7">vS</font></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">P</font> )
 
<b>by </b><i><a class="ref" href="valuat_1.html#T16" title="VALUAT_1:th.16">VALUAT_1:16</a></i>;<br/>
<a NAME="E4:94"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c9">vS2</font></span>)</span></span>)</span> <a href="valuat_1.html#K4" title="VALUAT_1:func.4">*'</a> <font color="Maroon" title="c5">ll</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">P</font> iff <span class="p1">(<span class="default"><a href="valuat_1.html#K8" title="VALUAT_1:func.8">Valid</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font></span>)</span>,<font color="Maroon" title="c3">J</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c9">vS2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 
<b>by </b><i><a class="ref" href="valuat_1.html#T16" title="VALUAT_1:th.16">VALUAT_1:16</a></i>;<br/>
<b>hence </b><a NAME="E5:94"/>
( <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c7">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font> iff <font color="Maroon" title="c3">J</font>,<font color="Maroon" title="c6">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c9">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c4">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Maroon" title="c5">ll</font> )
 <b>by </b><i><a class="txt" href="#E1:94"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="#E2:94"><i><font color="Green" title="E74">A2</font></i></a>, <a class="txt" href="#E3:94"><i><font color="Green" title="E75">A3</font></i></a>, <a class="ref" href="sublemma.html#T78" target="_self" title="SUBLEMMA:th.78">Th78</a>, <a class="ref" href="valuat_1.html#D12" title="VALUAT_1:def.12">VALUAT_1:def 12</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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