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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/>  for <font color="Olive" title="b2">F</font>, <font color="Olive" title="b3">G</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b2">F</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Olive" title="b3">G</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> for <font color="Olive" title="b4">FG</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b4">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; (  for <font color="Olive" title="b5">v</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> holds <br/> ( ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b5">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> implies <font color="Olive" title="b4">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b5">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Olive" title="b1">H</font> implies <font color="Olive" title="b4">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) ) ) holds <br/><font color="Olive" title="b4">FG</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</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">F</font>, <font color="Olive" title="b2">G</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">F</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Olive" title="b2">G</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> for <font color="Olive" title="b3">FG</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b3">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; (  for <font color="Olive" title="b4">v</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> holds <br/> ( ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b4">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> implies <font color="Olive" title="b3">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b4">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> implies <font color="Olive" title="b3">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) ) ) holds <br/><font color="Olive" title="b3">FG</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font>, <font color="Maroon" title="c4">G</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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"> ( <font color="Maroon" title="c3">F</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c4">G</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  for <font color="Olive" title="b1">FG</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; (  for <font color="Olive" title="b2">v</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> holds <br/> ( ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b2">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> implies <font color="Olive" title="b1">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b2">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> implies <font color="Olive" title="b1">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) ) ) holds <br/><font color="Olive" title="b1">FG</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font> )</span><br/>







<a NAME="E1:21"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T30" onmouseover="rs('enumset1/T30')" onmouseout="rh()">ENUMSET1:30</a></span>;<br/>
<a NAME="E2:21"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T2" onmouseover="rs('enumset1/T2')" onmouseout="rh()">ENUMSET1:2</a></span>;<br/>
<span class="kw">given </span><font color="Maroon" title="c5">H1</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><span class="kw"> such that </span><a NAME="E4:21"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">and </span><br/><a NAME="E5:21"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K12" title="ZF_LANG:func.12">Ex</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">H1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span>)</span>)</span>)</span>)</span>)
 <span class="kw">and </span><br/><a NAME="E6:21"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c1">M</font>)
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="zfmodel1.html#D3" onmouseover="rs('zfmodel1/D3')" onmouseout="rh()">ZFMODEL1:def 3</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"> (  not <font color="Maroon" title="c4">G</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font> or  not  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  for <font color="Olive" title="b1">FG</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; (  for <font color="Olive" title="b2">v</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> holds <br/> ( ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b2">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> implies <font color="Olive" title="b1">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b2">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> implies <font color="Olive" title="b1">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) ) ) holds <br/><font color="Olive" title="b1">FG</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c6">H2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><span class="kw"> such that </span><a NAME="E8:21"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c6">H2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">and </span><br/><a NAME="E9:21"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K12" title="ZF_LANG:func.12">Ex</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">H2</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span>)</span>)</span>)</span>)</span>)
 <span class="kw">and </span><br/><a NAME="E10:21"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c6">H2</font>,<font color="Maroon" title="c1">M</font>)
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="zfmodel1.html#D3" onmouseover="rs('zfmodel1/D3')" onmouseout="rh()">ZFMODEL1:def 3</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"> (  not  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or  for <font color="Olive" title="b1">FG</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; (  for <font color="Olive" title="b2">v</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> holds <br/> ( ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b2">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> implies <font color="Olive" title="b1">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b2">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> implies <font color="Olive" title="b1">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) ) ) holds <br/><font color="Olive" title="b1">FG</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E11:21"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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">  for <font color="Olive" title="b1">FG</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; (  for <font color="Olive" title="b2">v</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> holds <br/> ( ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b2">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> implies <font color="Olive" title="b1">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b2">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> implies <font color="Olive" title="b1">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) ) ) holds <br/><font color="Olive" title="b1">FG</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font></span><br/>

<a NAME="E12:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A10</font></span>: 
 not  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="zfmodel2.html#E13"><span class="lab"><font color="Green" title="E13">Lm3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">FG</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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"> (  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> &amp; (  for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> holds <br/> ( ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> implies <font color="Maroon" title="c7">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> implies <font color="Maroon" title="c7">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) ) ) implies <font color="Maroon" title="c7">FG</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E13:21"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">and </span><br/><a NAME="E14:21"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
 for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> holds <br/> ( ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> implies <font color="Maroon" title="c7">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> implies <font color="Maroon" title="c7">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span></span>)</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"> <font color="Maroon" title="c7">FG</font> <a href="zfmodel1.html#R1" title="ZFMODEL1:pred.1">is_definable_in</a> <font color="Maroon" title="c1">M</font></span><br/>

<a NAME="E15:21"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">FG</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c8">v</font> =  the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">FG</font> or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">FG</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="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:21_1"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">and </span><br/><a NAME="E4:21_1"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">b</font> = <font color="Maroon" title="c10">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a></span>;<br/>
<a NAME="E6:21_1"/><span class="lab"><font color="Green" title="E34">A15</font></span>: 
( <font color="Maroon" title="c1">M</font>, the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">b</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> or <font color="Maroon" title="c1">M</font>, the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">b</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T14" onmouseover="rs('zf_model/T14')" onmouseout="rh()">ZF_MODEL:14</a></span>;<br/>
<a NAME="E7:21_1"/>
<span class="p1">(<span class="default"> the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">b</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T128" onmouseover="rs('funct_7/T128')" onmouseout="rh()">FUNCT_7:128</a></span>;<br/>
<a NAME="E8:21_1"/><span class="kw">then </span>
( <font color="Maroon" title="c7">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font> or <font color="Maroon" title="c7">FG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c6">H2</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E6:21_1"><span class="lab"><font color="Green" title="E34">A15</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:21_1"/>
<font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_1"><span class="lab"><font color="Green" title="E33">A14</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/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">FG</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c9">I</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H1</font></span>)</span> <a href="zf_lang.html#K9" title="ZF_LANG:func.9">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c6">H2</font></span>)</span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="zfmodel1.html#D3" onmouseover="rs('zfmodel1/D3')" onmouseout="rh()">ZFMODEL1:def 3</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"> (  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K12" title="ZF_LANG:func.12">Ex</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span>)</span>)</span>)</span>)</span>) &amp; <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>) )</span><br/>

<a NAME="E17:21"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T60" onmouseover="rs('zf_lang1/T60')" onmouseout="rh()">ZF_LANG1:60</a></span>;<br/>
<a NAME="E18:21"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T61" onmouseover="rs('zf_lang1/T61')" onmouseout="rh()">ZF_LANG1:61</a></span>;<br/>
<a NAME="E19:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A17</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="xboole_1.html#T13" onmouseover="rs('xboole_1/T13')" onmouseout="rh()">XBOOLE_1:13</a></span>;<br/>
<a NAME="E20:21"/><span class="lab"><font color="Green" title="E34">A18</font></span>: 
 not  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c6">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="zfmodel2.html#E11"><span class="lab"><font color="Green" title="E11">Lm1</font></span></a>, <a class="txt" href="zfmodel2.html#E12"><span class="lab"><font color="Green" title="E12">Lm2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E21:21"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
 not  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="zfmodel2.html#E11"><span class="lab"><font color="Green" title="E11">Lm1</font></span></a>, <a class="txt" href="zfmodel2.html#E12"><span class="lab"><font color="Green" title="E12">Lm2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E22:21"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c6">H2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c6">H2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T61" onmouseover="rs('zf_lang1/T61')" onmouseout="rh()">ZF_LANG1:61</a></span>;<br/>
<a NAME="E23:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A20</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c6">H2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="xboole_1.html#T13" onmouseover="rs('xboole_1/T13')" onmouseout="rh()">XBOOLE_1:13</a></span>;<br/>
<a NAME="E24:21"/><span class="lab"><font color="Green" title="E37">A21</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H1</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c6">H2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T63" onmouseover="rs('zf_lang1/T63')" onmouseout="rh()">ZF_LANG1:63</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:21"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E23:21"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</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"> ( <font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K12" title="ZF_LANG:func.12">Ex</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span>)</span>)</span>)</span>)</span>) &amp; <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>) )</span><br/>

<a NAME="E26:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A22</font></span>: 
 not  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c9">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="zfmodel2.html#E11"><span class="lab"><font color="Green" title="E11">Lm1</font></span></a>, <a class="txt" href="zfmodel2.html#E12"><span class="lab"><font color="Green" title="E12">Lm2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E39">A23</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_2"><span class="kw">now </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">  for <font color="Olive" title="b1">v</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> holds  <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K12" title="ZF_LANG:func.12">Ex</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span>)</span>)</span>)</span>)</span>)</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">v</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</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="c1">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K12" title="ZF_LANG:func.12">Ex</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span>)</span>)</span>)</span>)</span>)</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_2_1"><span class="kw">now </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">  for <font color="Olive" title="b1">m3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>  ex <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> st <br/> for <font color="Olive" title="b3">m4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> holds <br/> ( ( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Olive" title="b1">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Olive" title="b3">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</font> implies <font color="Olive" title="b3">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> ) &amp; ( <font color="Olive" title="b3">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> implies <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Olive" title="b1">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Olive" title="b3">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</font> ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">m3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</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">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> st <br/> for <font color="Olive" title="b2">m4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> holds <br/> ( ( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Olive" title="b2">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</font> implies <font color="Olive" title="b2">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> ) &amp; ( <font color="Olive" title="b2">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> implies <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Olive" title="b2">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</font> ) )</span><br/><a NAME="E1:21_2_1"/>
<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K12" title="ZF_LANG:func.12">Ex</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">H1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span>)</span>)</span>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E23">A4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">m1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E3:21_2_1"/><span class="lab"><font color="Green" title="E40">A24</font></span>: 
 for <font color="Olive" title="b1">m4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> holds <br/> ( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Olive" title="b1">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c5">H1</font> iff <font color="Olive" title="b1">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="zfmodel2.html#T19" target="_self" onmouseover="rs('zfmodel2/T19')" onmouseout="rh()">Th19</a></span>;<br/><a NAME="E4:21_2_1"/>
<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K12" title="ZF_LANG:func.12">Ex</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">H2</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span>)</span>)</span>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">m2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E6:21_2_1"/><span class="lab"><font color="Green" title="E41">A25</font></span>: 
 for <font color="Olive" title="b1">m4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> holds <br/> ( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Olive" title="b1">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H2</font> iff <font color="Olive" title="b1">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:21"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="ref" href="zfmodel2.html#T19" target="_self" onmouseover="rs('zfmodel2/T19')" onmouseout="rh()">Th19</a></span>;<br/><a NAME="E7:21_2_1"/>
( (  not <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> ) or ( <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> &amp;  not <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T14" onmouseover="rs('zf_model/T14')" onmouseout="rh()">ZF_MODEL:14</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E9:21_2_1"/><span class="lab"><font color="Green" title="E42">A26</font></span>: 
( (  not <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">m1</font> ) or ( <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">m2</font> &amp;  not <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> ) )
 ;<br/><span class="kw">take </span><font color="Maroon" title="c15">m</font> = <font color="Maroon" title="c14">m</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">m4</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> holds <br/> ( ( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Olive" title="b1">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</font> implies <font color="Olive" title="b1">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</font> ) &amp; ( <font color="Olive" title="b1">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</font> implies <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Olive" title="b1">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c16">m4</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</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="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</font> implies <font color="Maroon" title="c16">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</font> ) &amp; ( <font color="Maroon" title="c16">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</font> implies <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</font> ) )</span><br/><span class="kw">thus </span><a NAME="E10:21_2_1"/>
( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</font> implies <font color="Maroon" title="c16">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</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="c16">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</font> implies <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_2_1_1"/>
<font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</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="c16">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</font></span><br/>

<a NAME="E2:21_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c6">H2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T17" onmouseover="rs('zf_model/T17')" onmouseout="rh()">ZF_MODEL:17</a></span>;<br/>
<a NAME="E3:21_2_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c5">H1</font> ) or ( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H2</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T15" onmouseover="rs('zf_model/T15')" onmouseout="rh()">ZF_MODEL:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21_2_1_1"/>
<font color="Maroon" title="c16">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E3:21_2_1"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="txt" href="#E6:21_2_1"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="txt" href="#E9:21_2_1"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="zfmodel2.html#T9" target="_self" onmouseover="rs('zfmodel2/T9')" onmouseout="rh()">Th9</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/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E11:21_2_1"/>
<font color="Maroon" title="c16">m4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">m</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="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</font></span><br/><a NAME="E12:21_2_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c5">H1</font> ) or ( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E3:21_2_1"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="txt" href="#E6:21_2_1"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="txt" href="#E9:21_2_1"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="zfmodel2.html#T9" target="_self" onmouseover="rs('zfmodel2/T9')" onmouseout="rh()">Th9</a></span>;<br/><a NAME="E13:21_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c6">H2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T15" onmouseover="rs('zf_model/T15')" onmouseout="rh()">ZF_MODEL:15</a></span>;<br/><span class="kw">hence </span><a NAME="E14:21_2_1"/>
<font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">m3</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<font color="Maroon" title="c16">m4</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T17" onmouseover="rs('zf_model/T17')" onmouseout="rh()">ZF_MODEL:17</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/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:21_2"/>
<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K12" title="ZF_LANG:func.12">Ex</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span>)</span>)</span>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="ref" href="zfmodel2.html#T19" target="_self" onmouseover="rs('zfmodel2/T19')" onmouseout="rh()">Th19</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E28:21"/><span class="lab"><font color="Green" title="E40">A27</font></span>: 
<font color="Maroon" title="c1">M</font> <a href="zf_model.html#R2" title="ZF_MODEL:pred.2">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K12" title="ZF_LANG:func.12">Ex</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span>)</span>)</span>)</span>)</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"> <font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_3"><span class="kw">now </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">  for <font color="Olive" title="b1">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> holds  <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">v</font> =  the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font>;<br/><span class="kw">let </span><font color="Maroon" title="c11">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</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="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span><br/><span class="kw">set </span><font color="Maroon" title="c12">m4</font> = <span class="p1">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font>;<br/><a NAME="E1:21_3"/><span class="lab"><font color="Green" title="E41">A28</font></span>: 
<font color="Maroon" title="c1">M</font>, the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K12" title="ZF_LANG:func.12">Ex</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c9">I</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span>)</span>)</span>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:21"><span class="lab"><font color="Green" title="E39">A23</font></span></a></span>;<br/><a NAME="E2:21_3"/>
 <a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> (<font color="Maroon" title="c9">I</font>, the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:21"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="txt" href="#E19:21"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E23:21"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="txt" href="#E28:21"><span class="lab"><font color="Green" title="E40">A27</font></span></a>, <a class="ref" href="zfmodel2.html#T11" target="_self" onmouseover="rs('zfmodel2/T11')" onmouseout="rh()">Th11</a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><a NAME="E3:21_3"/><span class="kw">then </span>
<font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"> the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">a</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c9">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:21"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="txt" href="#E1:21_3"><span class="lab"><font color="Green" title="E41">A28</font></span></a>, <a class="ref" href="zfmodel2.html#T10" target="_self" onmouseover="rs('zfmodel2/T10')" onmouseout="rh()">Th10</a></span>;<br/><a NAME="E4:21_3"/><span class="kw">then </span>
( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"> the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">a</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"> the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">a</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c6">H2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T17" onmouseover="rs('zf_model/T17')" onmouseout="rh()">ZF_MODEL:17</a></span>;<br/><a NAME="E5:21_3"/><span class="kw">then </span>
( ( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"> the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">a</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"> the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">a</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c5">H1</font> &amp; <font color="Maroon" title="c1">M</font>, the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K12" title="ZF_LANG:func.12">Ex</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">H1</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span>)</span>)</span>)</span>)</span>) &amp;  <a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c5">H1</font>,<font color="Maroon" title="c1">M</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> (<font color="Maroon" title="c5">H1</font>, the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font>) ) or ( <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"> the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">a</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c1">M</font>,<span class="p1">(<span class="default"> the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">a</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font></span>)</span>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H2</font> &amp; <font color="Maroon" title="c1">M</font>, the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K12" title="ZF_LANG:func.12">Ex</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p3">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">H2</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 4</span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p5">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span></span>)</span>)</span>)</span>)</span>)</span>) &amp;  <a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c6">H2</font>,<font color="Maroon" title="c1">M</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> (<font color="Maroon" title="c6">H2</font>, the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="zfmodel2.html#T11" target="_self" onmouseover="rs('zfmodel2/T11')" onmouseout="rh()">Th11</a>, <a class="ref" href="zf_model.html#T15" onmouseover="rs('zf_model/T15')" onmouseout="rh()">ZF_MODEL:15</a></span>;<br/><a NAME="E6:21_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A29</font></span>: 
( ( <font color="Maroon" title="c1">M</font>, the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">a</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c2">H</font> &amp; <span class="p1">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font> ) or ( <font color="Maroon" title="c1">M</font>, the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">a</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c2">H</font> &amp; <span class="p1">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E17:21"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E21:21"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="txt" href="#E20:21"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="ref" href="zfmodel2.html#T9" target="_self" onmouseover="rs('zfmodel2/T9')" onmouseout="rh()">Th9</a>, <a class="ref" href="zfmodel2.html#T10" target="_self" onmouseover="rs('zfmodel2/T10')" onmouseout="rh()">Th10</a></span>;<br/><a NAME="E7:21_3"/>
<span class="p1">(<span class="default"> the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c1">M</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span>,<font color="Maroon" title="c11">a</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T128" onmouseover="rs('funct_7/T128')" onmouseout="rh()">FUNCT_7:128</a></span>;<br/><span class="kw">hence </span><a NAME="E8:21_3"/>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="txt" href="#E6:21_3"><span class="lab"><font color="Green" title="E42">A29</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E30:21"/>
<font color="Maroon" title="c7">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zfmodel1.html#K2" title="ZFMODEL1:func.2">def_func</a> (<font color="Maroon" title="c9">I</font>,<font color="Maroon" title="c1">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" onmouseover="rs('funct_2/T63')" onmouseout="rh()">FUNCT_2:63</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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