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<b>deffunc </b><font color="Maroon">H<sub>7</sub></font>(  <a href="zf_lang.html#NM1">Variable</a>,  <a href="zf_lang.html#NM1">Variable</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>2</sub></font> </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>8</sub></font>(  <a href="zf_lang.html#NM1">Variable</a>,  <a href="zf_lang.html#NM1">Variable</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>2</sub></font> </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>9</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = <span class="p1">(<span class="default"><a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>10</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>11</sub></font>(  <a href="zf_lang.html#NM1">Variable</a>,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st ( for <font color="Olive">b<sub>4</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> ) holds <br/><font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>12</sub></font>(  <a href="zf_lang.html#NM2">ZF-formula</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="zf_model.html#K4" target="_self">St</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E1:13_1_1"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>12</sub></font>(<font color="Olive">b<sub>1</sub></font>) iff ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( ( for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">b<sub>5</sub></font></span>)</span>,<font color="Maroon">H<sub>7</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">b<sub>5</sub></font></span>)</span>,<font color="Maroon">H<sub>8</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> ) ) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> holds <br/>( ( <font color="Olive">b<sub>4</sub></font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  implies <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>7</sub></font>( <a href="zf_lang.html#K18">Var1</a> <font color="Olive">b<sub>4</sub></font>, <a href="zf_lang.html#K19">Var2</a> <font color="Olive">b<sub>4</sub></font>) ) &amp; ( <font color="Olive">b<sub>4</sub></font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  implies <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>8</sub></font>( <a href="zf_lang.html#K18">Var1</a> <font color="Olive">b<sub>4</sub></font>, <a href="zf_lang.html#K19">Var2</a> <font color="Olive">b<sub>4</sub></font>) ) &amp; ( <font color="Olive">b<sub>4</sub></font> is <a href="zf_lang.html#V4">negative</a> implies ex <font color="Olive">b<sub>6</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>9</sub></font>(<font color="Olive">b<sub>6</sub></font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> ) ) &amp; ( <font color="Olive">b<sub>4</sub></font> is <a href="zf_lang.html#V5">conjunctive</a> implies ex <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>10</sub></font>(<font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> ) ) &amp; ( <font color="Olive">b<sub>4</sub></font> is <a href="zf_lang.html#V6">universal</a> implies ex <font color="Olive">b<sub>6</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>11</sub></font>( <a href="zf_lang.html#K23">bound_in</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>6</sub></font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Olive">b<sub>4</sub></font></span>)</span>,<font color="Olive">b<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> ) ) ) ) ) )
 
<b>by </b><i><a class="ref" href="zf_model.html#D3" target="_self">Def3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> ;<br/><a NAME="E1:13_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:13_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:13_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E2:13_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> ;<br/><a NAME="E3:13_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 ;<br/>

<a NAME="E2:13_1_1_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:13_1_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:13_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div>
<a NAME="E3:13_1_1"/><b>then </b><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>7</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>8</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)] )
 
;<br/>
<div><i><font color="Green">E9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/><b>thus </b><a NAME="E2:13_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>9</sub></font>(<font color="Maroon">c<sub>3</sub></font>)]
 ;<br/></div><b>end;</b></div>
<a NAME="E5:13_1_1"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>10</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_1_1_3"/>
( <font color="Maroon">c<sub>3</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E3:13_1_1_3"/><i><font color="Green">E11</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T31">ZFMISC_1:31</a></i>;<br/>
<a NAME="E4:13_1_1_3"/>
<span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/>

<a NAME="E2:13_1_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="zf_model.html#E3:13_1_1_3"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1_1_3_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:13_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>10</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>)]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:13_1_1"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>11</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>)]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>



<b>assume </b><a NAME="E1:13_1_1_4"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st ( for <font color="Olive">b<sub>4</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> ) holds <br/><font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> ;<br/>
<a NAME="E2:13_1_1_4"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st ( for <font color="Olive">b<sub>4</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> ) holds <br/><font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st ( for <font color="Olive">b<sub>4</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> ) holds <br/><font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) </span>}</span> 
 ;<br/>

<a NAME="E2:13_1_1_4_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>4</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>2</sub></font> st ( for <font color="Olive">b<sub>5</sub></font> being  <a href="zf_lang.html#NM1">Variable</a> st <font color="Olive">b<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> ) holds <br/><font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )
 
;<br/>
<b>hence </b><a NAME="E3:13_1_1_4_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K3" target="_self">VAL</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:13_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>11</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>)]
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E7:13_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H<sub>12</sub></font>(<font color="Maroon">c<sub>1</sub></font>)]
 <b>from </b><i><a class="ref" href="zf_model.html#S4" target="_self">ZF_MODEL:sch 4</a>(<a class="txt" href="zf_model.html#E1:13_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="zf_model.html#E3:13_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="zf_model.html#E4:13_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="zf_model.html#E5:13_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="zf_model.html#E6:13_1_1"><i><font color="Green">E11</font></i></a>);</i><br/>


</div>
