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<b>set </b><font color="Maroon">S</font> =  <a href="circcomb.html#K6" target="_self">1GateCircStr</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>,<font color="Maroon">x</font>;<br/>
<b>let </b><font color="Maroon">A1</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of  <a href="circcomb.html#K6" target="_self">1GateCircStr</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>,<font color="Maroon">x</font>, <font color="Maroon">A2</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of  <a href="circcomb.html#K6" target="_self">1GateCircStr</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>,<font color="Maroon">x</font>;<br/>

<b>assume </b><a NAME="E1:89_1_2"/><i><font color="Green">E108</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">X</font></span>)</span> <a href="circcomb.html#K1" target="_self">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">X</font></span>)</span> <a href="circcomb.html#K1" target="_self">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; not <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> )
 ;<br/>

<a NAME="E2:89_1_2"/>
the <a href="msualg_1.html#U1">OperSymbols</a> of <span class="p1">(<span class="default"><a href="circcomb.html#K6" target="_self">1GateCircStr</a> <font color="Maroon">p</font>,<font color="Maroon">f</font>,<font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:89_1_2"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:89_1_2"/><b>then </b>
( the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="grfunc_1.html#T18">GRFUNC_1:18</a></i>;<br/>
<b>hence </b><a NAME="E5:89_1_2"/>
contradiction
 <b>by </b><i/>;<br/>


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