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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:60"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  : ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="cqc_lang.html#K7">CQC-WFF</a> ,<a href="cqc_the1.html#K2" target="_self">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2">:]</a></span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="cqc_the1.html#R2" target="_self">is_a_proof_wrt</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="cqc_the1.html#K3" target="_self">Effect</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 ;<br/>

<b>hence </b><a NAME="E2:60"/>
 <a href="cqc_lang.html#K9">VERUM</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E76"><i><font color="Green">Lemma35</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cqc_the1.html#D1" target="_self">CQC_THE1:def 1</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="qc_lang1.html#NM8">QC-formula</a>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/>











<b>thus </b><a NAME="E3:60"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="cqc_the1.html#E77"><i><font color="Green">Lemma36</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:60"/>
<font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="cqc_the1.html#E78"><i><font color="Green">Lemma37</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:60"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="cqc_the1.html#E79"><i><font color="Green">Lemma38</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:60"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="cqc_the1.html#E80"><i><font color="Green">Lemma39</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:60"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="cqc_the1.html#E81"><i><font color="Green">Lemma40</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:60"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="cqc_the1.html#E82"><i><font color="Green">Lemma41</font></i></a></i>;<br/>

<b>thus </b><a NAME="E9:60"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="cqc_the1.html#E83"><i><font color="Green">Lemma42</font></i></a></i>;<br/>

<b>thus </b><a NAME="E10:60"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="cqc_lang.html#K7">CQC-WFF</a>  &amp; not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K23">still_not-bound_in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cqc_the1.html#E1:60"><i><font color="Green">E45</font></i></a>, <a class="txt" href="cqc_the1.html#E84"><i><font color="Green">Lemma43</font></i></a></i>;<br/>


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