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<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:4_1_2"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font>, <a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:4_1_2"/><i><font color="Green">E5</font></i>: 
 <a href="circuit2.html#K2" target="_self">Fix_inp</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:4_1_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font>, <a href="msafree2.html#K6">FreeEnv</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:4_1_2"/><i><font color="Green">E7</font></i>: 
 <a href="circuit2.html#K2" target="_self">Fix_inp</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_1_2_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/><b>assume </b><a NAME="E2:4_1_2_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="circuit2.html#E1:4_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><a href="circuit2.html#K2" target="_self">Fix_inp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><a NAME="E5:4_1_2_1"/><i><font color="Green">E10</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="circuit2.html#E1:4_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/><a NAME="E6:4_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:4_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><a NAME="E7:4_1_2_1"/><b>then </b><i><font color="Green">E11</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E5:4_1_2_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:4_1_2_1"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><a href="circuit2.html#K2" target="_self">Fix_inp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E4:4_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="circuit2.html#E1:4_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><a NAME="E9:4_1_2_1"/><i><font color="Green">E13</font></i>: 
<span class="p1">(<span class="default"><a href="circuit2.html#K2" target="_self">Fix_inp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E2:4_1_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="circuit2.html#E1:4_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="circuit2.html#E2:4_1_2_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<font color="Maroon">c<sub>10</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="circuit2.html#E1:4_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="msafree.html#D1">MSAFREE:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="circuit2.html#K2" target="_self">Fix_inp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="circuit2.html#E7:4_1_2_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="circuit2.html#E8:4_1_2_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="circuit2.html#E7:4_1_2_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="circuit2.html#E9:4_1_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="grfunc_1.html#T64">GRFUNC_1:64</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E7:4_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msafree.html#D1">MSAFREE:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:4_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="circuit2.html#E1:4_1_2"><i><font color="Green">E4</font></i></a>, <a class="txt" href="circuit2.html#E3:4_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="extens_1.html#T18">EXTENS_1:18</a></i>;<br/>


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