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<span class="kw">let </span><font color="Maroon" title="c1">Al</font> be    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-alphabet</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">  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" title="c1">Al</font><br/>  for <font color="Olive" title="b2">x</font> being   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon" title="c1">Al</font>  st (  for <font color="Olive" title="b3">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font>  ex <font color="Olive" title="b4">S</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font> st <br/>( <font color="Olive" title="b4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">Sub</font> ) ) holds <br/> for <font color="Olive" title="b3">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font>  ex <font color="Olive" title="b4">S</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font> st <br/>( <font color="Olive" title="b4">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">p</font>) &amp; <font color="Olive" title="b4">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">Sub</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" title="c1">Al</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">x</font> being   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon" title="c1">Al</font>  st (  for <font color="Olive" title="b2">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font>  ex <font color="Olive" title="b3">S</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font> st <br/>( <font color="Olive" title="b3">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b3">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Sub</font> ) ) holds <br/> for <font color="Olive" title="b2">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font>  ex <font color="Olive" title="b3">S</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font> st <br/>( <font color="Olive" title="b3">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">p</font>) &amp; <font color="Olive" title="b3">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Sub</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon" title="c1">Al</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">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font>  ex <font color="Olive" title="b2">S</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font> st <br/>( <font color="Olive" title="b2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b2">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Sub</font> ) ) implies  for <font color="Olive" title="b1">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font>  ex <font color="Olive" title="b2">S</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font> st <br/>( <font color="Olive" title="b2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>) &amp; <font color="Olive" title="b2">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Sub</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:12"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 for <font color="Olive" title="b1">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font>  ex <font color="Olive" title="b2">S</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font> st <br/>( <font color="Olive" title="b2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b2">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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">Sub</font> being   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font>  ex <font color="Olive" title="b2">S</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font> st <br/>( <font color="Olive" title="b2">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>) &amp; <font color="Olive" title="b2">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Sub</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">Sub</font> be   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</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">  ex <font color="Olive" title="b1">S</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font> st <br/>( <font color="Olive" title="b1">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>) &amp; <font color="Olive" title="b1">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Sub</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">Sub1</font> = <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c4">Sub</font>)</span>)</span></span>)</span> <a href="sublemma.html#K14" title="SUBLEMMA:func.14">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c4">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span></span>)</span></span>)</span>;<br/>
<a NAME="E2:12"/>
( <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c4">Sub</font>)</span>)</span></span>)</span> <a href="sublemma.html#K14" title="SUBLEMMA:func.14">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c4">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span></span>)</span></span>)</span> is   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font> iff <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c4">Sub</font>)</span>)</span></span>)</span> <a href="sublemma.html#K14" title="SUBLEMMA:func.14">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c4">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span></span>)</span></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<span class="p1">(<span class="default"><a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">Al</font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">Al</font></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D1" onmouseover="rs('substut1/D1')" onmouseout="rh()">SUBSTUT1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">Sub1</font> = <span class="p1">(<span class="default"><a href="substut1.html#K2" title="SUBSTUT1:func.2">@</a> <span class="p2">(<span class="default"><a href="substut1.html#K7" title="SUBSTUT1:func.7">RestrictSub</a> (<font color="Maroon" title="c3">x</font>,<span class="p3">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c4">Sub</font>)</span>)</span></span>)</span> <a href="sublemma.html#K14" title="SUBLEMMA:func.14">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="sublemma.html#K12" title="SUBLEMMA:func.12">|</a> <span class="p2">(<span class="default"><a href="substut1.html#K35" title="SUBSTUT1:func.35">S_Bound</a> <span class="p3"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c4">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span></span>)</span></span>)</span> as   <a href="substut1.html#NM1" title="SUBSTUT1:NM.1">CQC_Substitution</a> of <font color="Maroon" title="c1">Al</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#T45" onmouseover="rs('partfun1/T45')" onmouseout="rh()">PARTFUN1:45</a></span>;<br/>
<a NAME="E4:12"/>
 ex <font color="Olive" title="b1">S</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font> st <br/>( <font color="Olive" title="b1">S</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Olive" title="b1">S</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Sub1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E12">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">S1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substut1.html#K38" title="SUBSTUT1:func.38">CQC-Sub-WFF</a> <font color="Maroon" title="c1">Al</font><span class="kw"> such that </span><br/><a NAME="E6:12"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c7">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="substut2.html#K1" title="SUBSTUT2:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>,<font color="Maroon" title="c4">Sub</font></span><a href="substut2.html#K1" title="SUBSTUT2:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="substut2.html#T10" target="_self" onmouseover="rs('substut2/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">S1</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" title="c7">S1</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>) &amp; <font color="Maroon" title="c7">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Sub</font> )</span><br/>

<span class="kw">thus </span><a NAME="E7:12"/>
( <font color="Maroon" title="c7">S1</font> <a href="sublemma.html#K2" title="SUBLEMMA:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">p</font>) &amp; <font color="Maroon" title="c7">S1</font> <a href="substut1.html#K19" title="SUBSTUT1:func.19">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Sub</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E13">A2</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/>


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