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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()] <span class="kw">means </span>(  <a href="cqc_sim1.html#K7" title="CQC_SIM1:func.7">QuantNbr</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon">P<sub>1</sub></font>[$1] );<br/>
<a NAME="E1:38_1"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">p</font>)]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="38_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">p</font>)]</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)] )</span><br/>



<span class="kw">assume </span><a NAME="E1:38_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)]</span><br/>

<span class="kw">assume </span><a NAME="E2:38_1_1"/>
 <a href="cqc_sim1.html#K7" title="CQC_SIM1:func.7">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[ <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)]</span><br/>

<a NAME="E3:38_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K7" title="CQC_SIM1:func.7">QuantNbr</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_sim1.html#T18" onmouseover="rs('cqc_sim1/T18')" onmouseout="rh()">CQC_SIM1:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:38_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:38_1"/>
 for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">q</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Olive" title="b2">q</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="38_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">q</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c2">q</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:38_1_2"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">q</font>] )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c2">q</font>]</span><br/>

<span class="kw">assume </span><a NAME="E2:38_1_2"/>
 <a href="cqc_sim1.html#K7" title="CQC_SIM1:func.7">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c2">q</font>]</span><br/>

<a NAME="E3:38_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K7" title="CQC_SIM1:func.7">QuantNbr</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K7" title="CQC_SIM1:func.7">QuantNbr</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_sim1.html#T17" onmouseover="rs('cqc_sim1/T17')" onmouseout="rh()">CQC_SIM1:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:38_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c2">q</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E1:38_1_2"><span class="lab"><font color="Green" title="E23">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:38_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A4</font></span>: 
 for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b3">x</font> being   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b4">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b5">l</font> being   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Olive" title="b4">k</font>,<font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b6">P</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Olive" title="b4">k</font>,<font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b6">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Olive" title="b5">l</font>] &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Olive" title="b1">r</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Olive" title="b2">s</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">r</font>)] ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E1:38_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="cqc_sim1.html#T16" onmouseover="rs('cqc_sim1/T16')" onmouseout="rh()">CQC_SIM1:16</a></span>;<br/>
<a NAME="E4:38_1"/>
 for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="cqc_lang.html#S1" onmouseover="rs('cqc_lang/S1')" onmouseout="rh()">CQC_LANG:sch 1</a>(<a class="txt" href="#E3:38_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:38_1"/>
 for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()  st  <a href="cqc_sim1.html#K7" title="CQC_SIM1:func.7">QuantNbr</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
