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<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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="cqc_the3.html#R5" title="CQC_THE3:pred.5">|-|</a> <font color="Maroon" title="c2">q</font> iff <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="cqc_the3.html#R4" title="CQC_THE3:pred.4">|-|</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>



<div><span class="lab"><font color="Green" title="E30">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:32_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="cqc_the3.html#R4" title="CQC_THE3:pred.4">|-|</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p</font> <a href="cqc_the3.html#R5" title="CQC_THE3:pred.5">|-|</a> <font color="Maroon" title="c2">q</font></span><br/><a NAME="E2:32_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="cqc_the3.html#R2" title="CQC_THE3:pred.2">|-</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the3.html#T18" target="_self" title="CQC_THE3:th.18">Th18</a></span>;<br/><a NAME="E3:32_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c2">q</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the3.html#T11" target="_self" title="CQC_THE3:th.11">Th11</a></span>;<br/><a NAME="E4:32_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="cqc_the3.html#R2" title="CQC_THE3:pred.2">|-</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="cqc_the3.html#T18" target="_self" title="CQC_THE3:th.18">Th18</a></span>;<br/><a NAME="E5:32_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c2">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the3.html#T11" target="_self" title="CQC_THE3:th.11">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E6:32_1"/>
<font color="Maroon" title="c1">p</font> <a href="cqc_the3.html#R5" title="CQC_THE3:pred.5">|-|</a> <font color="Maroon" title="c2">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="cqc_the3.html#D5" target="_self" title="CQC_THE3:def.5">Def5</a></span>; <a class="txt" 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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:32_2"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="cqc_the3.html#R5" title="CQC_THE3:pred.5">|-|</a> <font color="Maroon" title="c2">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="cqc_the3.html#R4" title="CQC_THE3:pred.4">|-|</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><a NAME="E2:32_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">q</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the3.html#D5" target="_self" title="CQC_THE3:def.5">Def5</a></span>;<br/><a NAME="E3:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A5</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="cqc_the3.html#R2" title="CQC_THE3:pred.2">|-</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the3.html#T11" target="_self" title="CQC_THE3:th.11">Th11</a></span>;<br/><a NAME="E4:32_2"/>
<font color="Maroon" title="c1">p</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c2">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_2"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="cqc_the3.html#D5" target="_self" title="CQC_THE3:def.5">Def5</a></span>;<br/><a NAME="E5:32_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="cqc_the3.html#R2" title="CQC_THE3:pred.2">|-</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the3.html#T11" target="_self" title="CQC_THE3:th.11">Th11</a></span>;<br/><span class="kw">hence </span><a NAME="E6:32_2"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="cqc_the3.html#R4" title="CQC_THE3:pred.4">|-|</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_2"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="cqc_the3.html#T18" target="_self" title="CQC_THE3:th.18">Th18</a></span>; <a class="txt" 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>
<span class="kw">hence </span><a NAME="E3:32"/>
( <font color="Maroon" title="c1">p</font> <a href="cqc_the3.html#R5" title="CQC_THE3:pred.5">|-|</a> <font color="Maroon" title="c2">q</font> iff <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="cqc_the3.html#R4" title="CQC_THE3:pred.4">|-|</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>; <a class="txt" 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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