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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">H1</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/>  for <font color="Olive" title="b2">v1</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>  st  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="Olive" title="b1">H1</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b2">v1</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="Olive" title="b1">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>) holds <br/> ex <font color="Olive" title="b3">H2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> ex <font color="Olive" title="b4">v2</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> st <br/>( <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">H1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b3">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> &amp;  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="Olive" title="b3">H2</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b4">v2</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="Olive" title="b3">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#K1" title="ZFMODEL1:func.1">def_func'</a> (<font color="Olive" title="b1">H1</font>,<font color="Olive" title="b2">v1</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="Olive" title="b3">H2</font>,<font color="Olive" title="b4">v2</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">H1</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">v1</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>  st  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="c2">H1</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b1">v1</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="c2">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>) holds <br/> ex <font color="Olive" title="b2">H2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> ex <font color="Olive" title="b3">v2</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> st <br/>( <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b2">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> &amp;  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="Olive" title="b2">H2</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b3">v2</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="Olive" title="b2">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#K1" title="ZFMODEL1:func.1">def_func'</a> (<font color="Maroon" title="c2">H1</font>,<font color="Olive" title="b1">v1</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="Olive" title="b2">H2</font>,<font color="Olive" title="b3">v2</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">v1</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"> (  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="c2">H1</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">v1</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="c2">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>) implies  ex <font color="Olive" title="b1">H2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> ex <font color="Olive" title="b2">v2</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> st <br/>( <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">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> &amp;  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="Olive" title="b1">H2</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b2">v2</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="Olive" title="b1">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#K1" title="ZFMODEL1:func.1">def_func'</a> (<font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c3">v1</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="Olive" title="b1">H2</font>,<font color="Olive" title="b2">v2</font>) ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:18"/><span class="lab"><font color="Green" title="E19">A1</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="c2">H1</font>
 <span class="kw">and </span><br/><a NAME="E2:18"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c3">v1</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="c2">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>)
 ; <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">H2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> ex <font color="Olive" title="b2">v2</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> st <br/>( <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">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> &amp;  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="Olive" title="b1">H2</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b2">v2</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="Olive" title="b1">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#K1" title="ZFMODEL1:func.1">def_func'</a> (<font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c3">v1</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="Olive" title="b1">H2</font>,<font color="Olive" title="b2">v2</font>) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:18"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
 for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c2">H1</font> holds <br/><font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmodel2.html#T3" target="_self" onmouseover="rs('zfmodel2/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">H2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>, <font color="Maroon" title="c6">v2</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><span class="kw"> such that </span><br/><a NAME="E6:18"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
 for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp;  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c5">H2</font> &amp;  not <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 holds <br/><font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 <span class="kw">and </span><br/><a NAME="E7:18"/><span class="lab"><font color="Green" title="E23">A5</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">H2</font>
 <span class="kw">and </span><br/><a NAME="E8:18"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c6">v2</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">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="E9:18"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
 <a href="zfmodel1.html#K1" title="ZFMODEL1:func.1">def_func'</a> (<font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c3">v1</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">H2</font>,<font color="Maroon" title="c6">v2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="zfmodel2.html#T16" target="_self" onmouseover="rs('zfmodel2/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">H2</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">v2</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> st <br/>( <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">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> &amp;  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">H2</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Olive" title="b1">v2</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">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#K1" title="ZFMODEL1:func.1">def_func'</a> (<font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c3">v1</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">H2</font>,<font color="Olive" title="b1">v2</font>) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">v2</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"> ( <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">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> &amp;  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">H2</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c6">v2</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">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#K1" title="ZFMODEL1:func.1">def_func'</a> (<font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c3">v1</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">H2</font>,<font color="Maroon" title="c6">v2</font>) )</span><br/>

<span class="kw">thus </span><a NAME="E10:18"/>
<span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">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>
  <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_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">H2</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c6">v2</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">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#K1" title="ZFMODEL1:func.1">def_func'</a> (<font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c3">v1</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">H2</font>,<font color="Maroon" title="c6">v2</font>) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:18_1"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c5">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T151" onmouseover="rs('zf_lang1/T151')" onmouseout="rh()">ZF_LANG1:151</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">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="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">H1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">H2</font></span>)</span> or <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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><br/>

<a NAME="E2:18_1"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c2">H1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T151" onmouseover="rs('zf_lang1/T151')" onmouseout="rh()">ZF_LANG1:151</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:18_1"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">H1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</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">H2</font></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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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><br/>

<a NAME="E4:18_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">a</font> as   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:18_1"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K2" title="ZF_LANG:func.2">x.</a> <font color="Maroon" title="c9">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T77" onmouseover="rs('zf_lang1/T77')" onmouseout="rh()">ZF_LANG1:77</a></span>;<br/>
<a NAME="E8:18_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c2">H1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E9:18_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E6:18"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E4:18_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E7:18_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:18_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="#E7:18_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</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">thus </span><a NAME="E11:18"/>
(  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">H2</font> &amp; <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c6">v2</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">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#K1" title="ZFMODEL1:func.1">def_func'</a> (<font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c3">v1</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">H2</font>,<font color="Maroon" title="c6">v2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:18"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E25">A7</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>
