<?xml version="1.0"?>
<div class="add">

<b>thus </b><a NAME="E1:2_1"/>
 ex <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ex <font color="Olive">F</font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>() &amp; ( for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <br/> for <font color="Olive">d1</font>, <font color="Olive">d2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( ( <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11">VERUM</a>  implies <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">p</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V3">negative</a> &amp; <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Olive">p</font></span>)</span> implies <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">d1</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Olive">p</font></span>)</span> &amp; <font color="Olive">d2</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Olive">p</font></span>)</span> implies <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive">d1</font>,<font color="Olive">d2</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V5">universal</a> &amp; <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Olive">p</font></span>)</span> implies <font color="Olive">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">p</font>,<font color="Olive">d1</font>) ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E2:2_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() &amp; ( for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  holds <br/> ( ( <font color="Olive">p</font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">p</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V3">negative</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Olive">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V4">conjunctive</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Olive">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Olive">p</font></span>)</span></span>)</span>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V5">universal</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">p</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Olive">p</font></span>)</span></span>)</span>) ) ) ) )
 <b>from </b><i><a class="ref" href="qc_lang1.html#S3">QC_LANG1:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>()
;<br/>

<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E3:2_1_1"/>
( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>() &amp; ( for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <br/> for <font color="Olive">d1</font>, <font color="Olive">d2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( ( <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11">VERUM</a>  implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">p</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V3">negative</a> &amp; <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Olive">p</font></span>)</span> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">d1</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Olive">p</font></span>)</span> &amp; <font color="Olive">d2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Olive">p</font></span>)</span> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive">d1</font>,<font color="Olive">d2</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V5">universal</a> &amp; <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Olive">p</font></span>)</span> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">p</font>,<font color="Olive">d1</font>) ) ) ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">x2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>given </b><font color="Maroon">F1</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><a NAME="E2:2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>and </b><br/><a NAME="E3:2_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <br/> for <font color="Olive">d1</font>, <font color="Olive">d2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( ( <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11">VERUM</a>  implies <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">p</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V3">negative</a> &amp; <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Olive">p</font></span>)</span> implies <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">d1</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Olive">p</font></span>)</span> &amp; <font color="Olive">d2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Olive">p</font></span>)</span> implies <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive">d1</font>,<font color="Olive">d2</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V5">universal</a> &amp; <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Olive">p</font></span>)</span> implies <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">p</font>,<font color="Olive">d1</font>) ) )
 ;<br/>

<b>given </b><font color="Maroon">F2</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ,<font color="Maroon">F<sub>1</sub></font>()<b> such that </b><a NAME="E4:2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>and </b><br/><a NAME="E5:2_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <br/> for <font color="Olive">d1</font>, <font color="Olive">d2</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( ( <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11">VERUM</a>  implies <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">p</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V3">negative</a> &amp; <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Olive">p</font></span>)</span> implies <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive">d1</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Olive">p</font></span>)</span> &amp; <font color="Olive">d2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Olive">p</font></span>)</span> implies <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive">d1</font>,<font color="Olive">d2</font>) ) &amp; ( <font color="Olive">p</font> is <a href="qc_lang1.html#V5">universal</a> &amp; <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Olive">p</font></span>)</span> implies <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive">p</font>,<font color="Olive">d1</font>) ) )
 ;<br/>

<a NAME="E6:2_1"/>
<font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F2</font>
 
<b>from </b><i><a class="ref" href="qc_lang3.html#S1" target="_self">QC_LANG3:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E7:2_1"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, </i>;<br/>


</div>
