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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="msafree2.html#V2">Circuit-like</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>











<b>assume </b><a NAME="E1:76"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="circtrm1.html#R4" target="_self">form_embedding_of</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="circtrm1.html#R3" target="_self">preserves_inputs_of</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>



<b>assume </b><a NAME="E2:76"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E3:76"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="pua2mss1.html#R3">form_morphism_between</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="circtrm1.html#E1:76"><i><font color="Green">E52</font></i></a>, <a class="ref" href="circtrm1.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E4:76"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pua2mss1.html#D13">PUA2MSS1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="circtrm1.html#E3:76"><i><font color="Green">E54</font></i></a>, <a class="ref" href="circtrm1.html#T31" target="_self">Th31</a></i>;<br/>
<b>thus </b><a NAME="E6:76"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">c<sub>9</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="facirc_1.html#R1">is_stable_at</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a><br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a><br/>

<a NAME="E2:76_1"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="circtrm1.html#E1:76"><i><font color="Green">E52</font></i></a>, <a class="txt" href="circtrm1.html#E2:76"><i><font color="Green">E53</font></i></a>, <a class="ref" href="circtrm1.html#T48" target="_self">Th48</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="circtrm1.html#E4:76"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="circtrm1.html#E1:76_1"><i><font color="Green">E56</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="circtrm1.html#E2:76"><i><font color="Green">E53</font></i></a>, <a class="txt" href="circtrm1.html#E4:76"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><br/>


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

<b>assume </b><a NAME="E7:76"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a><br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="facirc_1.html#D9">FACIRC_1:def 9</a><br/>

<a NAME="E8:76"/>
 <a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="circtrm1.html#E1:76"><i><font color="Green">E52</font></i></a>, <a class="txt" href="circtrm1.html#E2:76"><i><font color="Green">E53</font></i></a>, <a class="ref" href="circtrm1.html#T48" target="_self">Th48</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><a href="facirc_1.html#K9">Following</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="circtrm1.html#E4:76"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="circtrm1.html#E7:76"><i><font color="Green">E56</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="circtrm1.html#E2:76"><i><font color="Green">E53</font></i></a>, <a class="txt" href="circtrm1.html#E4:76"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><br/>


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