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<b>let </b><font color="Maroon" title="c1">p</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">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b2">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="b3">J</font> being   <a href="valuat_1.html#M1" title="VALUAT_1:mode.1">interpretation</a> of <font color="Olive" title="b2">A</font>  st ( 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="Olive" title="b2">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="Olive" title="b2">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> <font color="Maroon" title="c1">p</font> ) &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="b3">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="Maroon" title="c1">p</font> iff <font color="Olive" title="b3">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="Maroon" title="c1">p</font> ) ) holds <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="Olive" title="b2">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="Olive" title="b2">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"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">p</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="b3">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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">p</font> iff <font color="Olive" title="b3">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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">p</font> )</span><br/><b>let </b><font color="Maroon" title="c2">x</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"> 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> ) ) 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"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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> ) ) 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"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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> ) ) 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"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font> ) )</span><br/>







<b>assume </b><a NAME="E1:99"/><i><font color="Green" title="E78">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> )
 ; <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"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font> ) )</span><br/>







<b>assume </b><a NAME="E2:99"/><i><font color="Green" title="E79">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"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font> )</span><br/>

<b>set </b><font color="Maroon" title="c9">vS1'</font> = <font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">vS2'</font> = <font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<a NAME="E3:99"/><i><font color="Green" title="E80">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></i>;<br/>
<a NAME="E4:99"/><b>then </b><i><font color="Green" title="E81">A4</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E5:99"/><i><font color="Green" title="E82">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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> 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="sublemma.html#E2:99"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="sublemma.html#E3:99"><i><font color="Green" title="E80">A3</font></i></a>, <a class="ref" href="sublemma.html#T82" target="_self" title="SUBLEMMA:th.82">Th82</a></i>;<br/>
<a NAME="E6:99"/><i><font color="Green" title="E83">A6</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> <span class="p1">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font>
 
<b>by </b><i><a class="txt" href="sublemma.html#E2:99"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="sublemma.html#E3:99"><i><font color="Green" title="E80">A3</font></i></a>, <a class="ref" href="sublemma.html#T83" target="_self" title="SUBLEMMA:th.83">Th83</a></i>;<br/>
<a NAME="E7:99"/><i><font color="Green" title="E84">A7</font></i>: 
( ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) iff for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:99_1"/>
( ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) implies for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) implies for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:99_1_1"/><i><font color="Green" title="E85">A8</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font></span><br/>

<b>let </b><font color="Maroon" title="c11">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font></span><br/>

<a NAME="E2:99_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E4:99"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="sublemma.html#T59" target="_self" title="SUBLEMMA:th.59">Th59</a></i>;<br/>
<a NAME="E3:99_1_1"/><b>then </b>
( <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <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>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:99"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="sublemma.html#E5:99"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="sublemma.html#E6:99"><i><font color="Green" title="E83">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:99_1_1"/>
<font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:99_1_1"><i><font color="Green" title="E85">A8</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="E2:99_1"/>
( ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) implies for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</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="99_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:99_1_2"/><i><font color="Green" title="E85">A9</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font></span><br/>

<b>let </b><font color="Maroon" title="c11">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font></span><br/>

<a NAME="E2:99_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E4:99"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="sublemma.html#T59" target="_self" title="SUBLEMMA:th.59">Th59</a></i>;<br/>
<a NAME="E3:99_1_2"/><b>then </b>
( <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <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>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="txt" href="sublemma.html#E1:99"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="sublemma.html#E5:99"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="sublemma.html#E6:99"><i><font color="Green" title="E83">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:99_1_2"/>
<font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:99_1_2"><i><font color="Green" title="E85">A9</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>
<a NAME="E8:99"/><i><font color="Green" title="E85">A10</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="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></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="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></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span><br/>

<a NAME="E1:99_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E4:99"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="sublemma.html#T59" target="_self" title="SUBLEMMA:th.59">Th59</a></i>;<br/>
<a NAME="E2:99_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T36" title="FUNCT_4:th.36">FUNCT_4:36</a></i>;<br/>
<a NAME="E3:99_2"/><b>then </b><i><font color="Green" title="E86">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T15" title="FUNCT_4:th.15">FUNCT_4:15</a></i>;<br/>
<a NAME="E4:99_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E4:99"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="sublemma.html#T59" target="_self" title="SUBLEMMA:th.59">Th59</a></i>;<br/>
<a NAME="E5:99_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E3:99_2"><i><font color="Green" title="E86">A11</font></i></a>, <a class="ref" href="funct_4.html#T36" title="FUNCT_4:th.36">FUNCT_4:36</a></i>;<br/>
<a NAME="E6:99_2"/><b>then </b><i><font color="Green" title="E87">A12</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T15" title="FUNCT_4:th.15">FUNCT_4:15</a></i>;<br/>
<div><i><font color="Green" title="E88">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span></span><br/><a NAME="E2:99_2_1"/><b>then </b><i><font color="Green" title="E89">A14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">vS1</font>
 <b>by </b><i><a class="ref" href="sublemma.html#T2" target="_self" title="SUBLEMMA:th.2">Th2</a></i>;<br/><div><i><font color="Green" title="E90">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_2_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span></span><br/><a NAME="E2:99_2_1_1"/><b>then </b><i><font color="Green" title="E91">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vS2</font>
 <b>by </b><i><a class="ref" href="sublemma.html#T2" target="_self" title="SUBLEMMA:th.2">Th2</a></i>;<br/><a NAME="E3:99_2_1_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E4:99_2_1_1"/><b>then </b>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> )
 <b>by </b><i><a class="ref" href="sublemma.html#T59" target="_self" title="SUBLEMMA:th.59">Th59</a></i>;<br/><b>hence </b><a NAME="E5:99_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E6:99_2"><i><font color="Green" title="E87">A12</font></i></a>, <a class="txt" href="sublemma.html#E2:99_2_1"><i><font color="Green" title="E89">A14</font></i></a>, <a class="txt" href="sublemma.html#E2:99_2_1_1"><i><font color="Green" title="E91">A16</font></i></a>, <a class="ref" href="sublemma.html#T1" target="_self" title="SUBLEMMA:th.1">Th1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_2_1_2"/>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span></span><br/><a NAME="E2:99_2_1_2"/><b>then </b>
<font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>;<br/><a NAME="E3:99_2_1_2"/><b>then </b>
<font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">vS2</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/><a NAME="E4:99_2_1_2"/><b>then </b><i><font color="Green" title="E91">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vS2</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></i>;<br/><a NAME="E5:99_2_1_2"/><i><font color="Green" title="E92">A18</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E6:99_2_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E7:99_2_1_2"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="sublemma.html#T59" target="_self" title="SUBLEMMA:th.59">Th59</a></i>;<br/><b>hence </b><a NAME="E8:99_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E6:99_2"><i><font color="Green" title="E87">A12</font></i></a>, <a class="txt" href="sublemma.html#E2:99_2_1"><i><font color="Green" title="E89">A14</font></i></a>, <a class="txt" href="sublemma.html#E4:99_2_1_2"><i><font color="Green" title="E91">A17</font></i></a>, <a class="txt" href="sublemma.html#E5:99_2_1_2"><i><font color="Green" title="E92">A18</font></i></a>, <a class="ref" href="sublemma.html#T1" target="_self" title="SUBLEMMA:th.1">Th1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:99_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E3:99_2_1"><i><font color="Green" title="E90">A15</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_2_2"/>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span></span><br/><a NAME="E2:99_2_2"/><b>then </b>
<font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>;<br/><a NAME="E3:99_2_2"/><b>then </b><i><font color="Green" title="E89">A19</font></i>: 
<font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c7">vS1</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/><a NAME="E4:99_2_2"/><b>then </b><i><font color="Green" title="E90">A20</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">vS1</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></i>;<br/><a NAME="E5:99_2_2"/><i><font color="Green" title="E91">A21</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><div><i><font color="Green" title="E92">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_2_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span></span><br/><a NAME="E2:99_2_2_1"/><b>then </b><i><font color="Green" title="E93">A23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">vS2</font>
 <b>by </b><i><a class="ref" href="sublemma.html#T2" target="_self" title="SUBLEMMA:th.2">Th2</a></i>;<br/><a NAME="E3:99_2_2_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E4:99_2_2_1"/><b>then </b>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="sublemma.html#T59" target="_self" title="SUBLEMMA:th.59">Th59</a></i>;<br/><b>hence </b><a NAME="E5:99_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E6:99_2"><i><font color="Green" title="E87">A12</font></i></a>, <a class="txt" href="sublemma.html#E4:99_2_2"><i><font color="Green" title="E90">A20</font></i></a>, <a class="txt" href="sublemma.html#E5:99_2_2"><i><font color="Green" title="E91">A21</font></i></a>, <a class="txt" href="sublemma.html#E2:99_2_2_1"><i><font color="Green" title="E93">A23</font></i></a>, <a class="ref" href="sublemma.html#T1" target="_self" title="SUBLEMMA:th.1">Th1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_2_2_2"/>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span></span><br/><a NAME="E2:99_2_2_2"/><b>then </b>
<font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T65" title="ZFMISC_1:th.65">ZFMISC_1:65</a></i>;<br/><b>hence </b><a NAME="E3:99_2_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E6:99_2"><i><font color="Green" title="E87">A12</font></i></a>, <a class="txt" href="sublemma.html#E3:99_2_2"><i><font color="Green" title="E89">A19</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E8:99_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="sublemma.html#E6:99_2_2"><i><font color="Green" title="E92">A22</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E9:99_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E7:99_2"><i><font color="Green" title="E88">A13</font></i></a>, <a class="ref" href="funct_4.html#T15" title="FUNCT_4:th.15">FUNCT_4:15</a></i>;<br/>
<a NAME="E10:99_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c6">vS</font></span>)</span> <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="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T15" title="FUNCT_4:th.15">FUNCT_4:15</a></i>;<br/>
<a NAME="E11:99_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<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></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T15" title="FUNCT_4:th.15">FUNCT_4:15</a></i>;<br/>
<b>hence </b><a NAME="E12:99_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="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></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T15" title="FUNCT_4:th.15">FUNCT_4:15</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="E9:99"/>
( ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) iff for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="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></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:99_3"/>
( ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) implies for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="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></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="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></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) implies for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:99_3_1"/><i><font color="Green" title="E86">A24</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="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></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font></span><br/>

<b>let </b><font color="Maroon" title="c11">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="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></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font></span><br/>

<a NAME="E2:99_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="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></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E8:99"><i><font color="Green" title="E85">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:99_3_1"/>
<font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="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></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:99_3_1"><i><font color="Green" title="E86">A24</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="E2:99_3"/>
( ( for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="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></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font> ) implies for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</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="99_3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:99_3_2"/><i><font color="Green" title="E86">A25</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">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="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></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">A</font> holds  <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font></span><br/>

<b>let </b><font color="Maroon" title="c11">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> <font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font></span><br/>

<a NAME="E2:99_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="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></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="sublemma.html#E8:99"><i><font color="Green" title="E85">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:99_3_2"/>
<font color="Maroon" title="c4">J</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <font color="Maroon" title="c6">vS</font></span>)</span> <a href="sublemma.html#K1" title="SUBLEMMA:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="sublemma.html#K13" title="SUBLEMMA:func.13">|</a> <font color="Maroon" title="c11">a</font></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">vS1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">vS1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="sublemma.html#K15" title="SUBLEMMA:func.15">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">vS2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c8">vS2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span></span>)</span> <a href="valuat_1.html#R1" title="VALUAT_1:pred.1">|=</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="sublemma.html#E1:99_3_2"><i><font color="Green" title="E86">A25</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="E10:99"/>
( <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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</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>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">p</font> )
 <b>by </b><i><a class="txt" href="sublemma.html#E7:99"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="sublemma.html#T51" target="_self" title="SUBLEMMA:th.51">Th51</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>
