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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</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"> 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>  st ( 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="Olive" title="b1">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="Olive" title="b1">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> <font color="Maroon" title="c1">p</font> ) &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="Olive" title="b2">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="Maroon" title="c1">p</font> iff <font color="Olive" title="b2">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="Maroon" title="c1">p</font> ) ) &amp; ( 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="Olive" title="b1">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="Olive" title="b1">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> <font color="Maroon" title="c2">q</font> ) &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="Olive" title="b2">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="Maroon" title="c2">q</font> iff <font color="Olive" title="b2">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="Maroon" title="c2">q</font> ) ) holds <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="Olive" title="b1">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="Olive" title="b1">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="Maroon" title="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</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="Olive" title="b2">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="Maroon" title="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> iff <font color="Olive" title="b2">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="Maroon" title="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> )</span><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> ; <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>  st ( 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><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="c3">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> <font color="Maroon" title="c1">p</font> ) &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="Olive" title="b1">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="c1">p</font> iff <font color="Olive" title="b1">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="c1">p</font> ) ) &amp; ( 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><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="c3">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> <font color="Maroon" title="c2">q</font> ) &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="Olive" title="b1">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="c2">q</font> iff <font color="Olive" title="b1">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="c2">q</font> ) ) holds <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><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="c3">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</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="Olive" title="b1">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> iff <font color="Olive" title="b1">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</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><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="c3">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> <font color="Maroon" title="c1">p</font> ) &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="c4">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="c1">p</font> iff <font color="Maroon" title="c4">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="c1">p</font> ) ) &amp; ( 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><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="c3">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> <font color="Maroon" title="c2">q</font> ) &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="c4">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="c2">q</font> iff <font color="Maroon" title="c4">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="c2">q</font> ) ) implies 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><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="c3">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</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="c4">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> iff <font color="Maroon" title="c4">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> ) )</span><br/>







<b>assume </b><a NAME="E1:96"/><i><font color="Green" title="E75">A1</font></i>: 
( ( 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><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="c3">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> <font color="Maroon" title="c1">p</font> ) &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="c4">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="c1">p</font> iff <font color="Maroon" title="c4">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="c1">p</font> ) ) &amp; ( 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><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="c3">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> <font color="Maroon" title="c2">q</font> ) &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="c4">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="c2">q</font> iff <font color="Maroon" title="c4">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="c2">q</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><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="c3">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</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="c4">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> iff <font color="Maroon" title="c4">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</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"> 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="c3">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</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="c5">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="c4">J</font>,<font color="Maroon" title="c5">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> iff <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> )</span><br/><b>let </b><font color="Maroon" title="c6">vS</font>, <font color="Maroon" title="c7">vS1</font>, <font color="Maroon" title="c8">vS2</font> be   <a href="sublemma.html#NM1" title="SUBLEMMA:NM.1">Val_Sub</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">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="c7">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</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="c8">vS2</font> holds <br/><font color="Maroon" title="c8">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="c5">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="c6">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="c8">vS2</font> implies ( <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> iff <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> ) )</span><br/>







<b>assume </b><a NAME="E2:96"/><i><font color="Green" title="E76">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="c7">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</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="c8">vS2</font> holds <br/><font color="Maroon" title="c8">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="c5">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="c6">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="c8">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="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> iff <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> )</span><br/>

<a NAME="E3:96"/><i><font color="Green" title="E77">A3</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="c7">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> <font color="Maroon" title="c1">p</font> &amp; 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> <font color="Maroon" title="c2">q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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="c9">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="c7">vS1</font> implies ( not <font color="Maroon" title="c9">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> <font color="Maroon" title="c1">p</font> &amp; not <font color="Maroon" title="c9">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> <font color="Maroon" title="c2">q</font> ) )</span><br/>

<b>assume </b><a NAME="E1:96_1"/>
<font color="Maroon" title="c9">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="c7">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="c9">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> <font color="Maroon" title="c1">p</font> &amp; not <font color="Maroon" title="c9">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> <font color="Maroon" title="c2">q</font> )</span><br/>

<a NAME="E2:96_1"/><b>then </b>
not <font color="Maroon" title="c9">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="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:96"><i><font color="Green" title="E76">A2</font></i></a></i>;<br/>
<a NAME="E3:96_1"/><b>then </b>
not <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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="c2">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T14" title="QC_LANG3:th.14">QC_LANG3:14</a></i>;<br/>
<b>hence </b><a NAME="E4:96_1"/>
( not <font color="Maroon" title="c9">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> <font color="Maroon" title="c1">p</font> &amp; not <font color="Maroon" title="c9">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> <font color="Maroon" title="c2">q</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0: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="E4:96"/>
( <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</font> iff ( <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:96_2"/>
( <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</font> implies ( <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</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="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</font> implies ( <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:96_2_1"/><i><font color="Green" title="E78">A4</font></i>: 
( <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</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="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</font> )</span><br/>

<a NAME="E2:96_2_1"/><i><font color="Green" title="E79">A5</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="c7">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> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="#E3:96"><i><font color="Green" title="E77">A3</font></i></a></i>;<br/>
<a NAME="E3:96_2_1"/>
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="c7">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> <font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="txt" href="#E3:96"><i><font color="Green" title="E77">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:96_2_1"/>
( <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</font> )
 <b>by </b><i><a class="txt" href="#E1:96"><i><font color="Green" title="E75">A1</font></i></a>, <a class="txt" href="#E2:96"><i><font color="Green" title="E76">A2</font></i></a>, <a class="txt" href="#E1:96_2_1"><i><font color="Green" title="E78">A4</font></i></a>, <a class="txt" href="#E2:96_2_1"><i><font color="Green" title="E79">A5</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>thus </b><a NAME="E2:96_2"/>
( <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</font> implies ( <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:96_2_2"/><i><font color="Green" title="E78">A6</font></i>: 
( <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</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="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</font> )</span><br/>

<a NAME="E2:96_2_2"/><i><font color="Green" title="E79">A7</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="c7">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> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="#E3:96"><i><font color="Green" title="E77">A3</font></i></a></i>;<br/>
<a NAME="E3:96_2_2"/>
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="c7">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> <font color="Maroon" title="c2">q</font>
 
<b>by </b><i><a class="txt" href="#E3:96"><i><font color="Green" title="E77">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:96_2_2"/>
( <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c2">q</font> )
 <b>by </b><i><a class="txt" href="#E1:96"><i><font color="Green" title="E75">A1</font></i></a>, <a class="txt" href="#E2:96"><i><font color="Green" title="E76">A2</font></i></a>, <a class="txt" href="#E1:96_2_2"><i><font color="Green" title="E78">A6</font></i></a>, <a class="txt" href="#E2:96_2_2"><i><font color="Green" title="E79">A7</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>
<b>hence </b><a NAME="E5:96"/>
( <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> iff <font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">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="c6">vS</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c2">q</font> )
 <b>by </b><i><a class="ref" href="valuat_1.html#T29" title="VALUAT_1:th.29">VALUAT_1:29</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>
