<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c1">I</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<a NAME="E1:7_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon">F<sub>2</sub></font>() is <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>()
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">i</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="pboole.html#D13" onmouseover="rs('pboole/D13')" onmouseout="rh()">PBOOLE:def 13</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="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() or  not <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_1_1"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></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 <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">fi</font> = <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D15" onmouseover="rs('pboole/D15')" onmouseout="rh()">PBOOLE:def 15</a></span>;<br/>
<a NAME="E3:7_1_1"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">fi</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="pboole.html#D20" onmouseover="rs('pboole/D20')" onmouseout="rh()">PBOOLE:def 20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">Xi</font> = <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> as   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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>;<br/>
<a NAME="E5:7_1_1"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon" title="c4">Xi</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c4">Xi</font>
 
;<br/>
<a NAME="E6:7_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Xi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:7_1_1"/>
 not <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7_1_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E3:7_1_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="relat_1.html#T118" onmouseover="rs('relat_1/T118')" onmouseout="rh()">RELAT_1:118</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>
<a NAME="E2:7_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon">F<sub>2</sub></font>() is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">i</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="pboole.html#D2" onmouseover="rs('pboole/D2')" onmouseout="rh()">PBOOLE:def 2</a>,<a class="ref" href="pboole.html#D18" onmouseover="rs('pboole/D18')" onmouseout="rh()">PBOOLE:def 18</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="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() or <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_1_2"/><span class="lab"><font color="Green" title="E7">A8</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></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"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">fi</font> = <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span>,<span class="p1">(<span class="default"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D15" onmouseover="rs('pboole/D15')" onmouseout="rh()">PBOOLE:def 15</a></span>;<br/>
<a NAME="E3:7_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">fi</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1_2"><span class="lab"><font color="Green" title="E7">A8</font></span></a>, <a class="ref" href="pboole.html#D20" onmouseover="rs('pboole/D20')" onmouseout="rh()">PBOOLE:def 20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:7_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">Gen</font> = <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon">F<sub>2</sub></font>() as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>() <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon">F<sub>3</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">AA</font> =  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font>;<br/>
<a NAME="E4:7_1"/><span class="lab"><font color="Green" title="E7">A9</font></span>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font></span>)</span>
 
;<br/>
<a NAME="E5:7_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()
 
;<br/>
<a NAME="E6:7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A10</font></span>: 
 <a href="funct_6.html#K1" title="FUNCT_6:func.1">doms</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#T17" onmouseover="rs('mssubfam/T17')" onmouseout="rh()">MSSUBFAM:17</a></span>;<br/>
<a NAME="E7:7_1"/><span class="kw">then </span>
 <a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="equation.html#T13" onmouseover="rs('equation/T13')" onmouseout="rh()">EQUATION:13</a></span>;<br/>
<a NAME="E8:7_1"/><span class="kw">then </span>
 <a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <font color="Maroon">F<sub>4</sub></font>() is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D17" onmouseover="rs('msualg_2/D17')" onmouseout="rh()">MSUALG_2:def 17</a></span>;<br/>
<a NAME="E9:7_1"/><span class="kw">then </span>
 <a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" onmouseover="rs('pboole/D18')" onmouseout="rh()">PBOOLE:def 18</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">iN</font> = <font color="Maroon">F<sub>4</sub></font>() as    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(), the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7_1"><span class="lab"><font color="Green" title="E7">A9</font></span></a>, <a class="txt" href="#E6:7_1"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="ref" href="equation.html#T4" onmouseover="rs('equation/T4')" onmouseout="rh()">EQUATION:4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">IN</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>3</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E12:7_1"/><span class="lab"><font color="Green" title="E9">A11</font></span>: 
<font color="Maroon" title="c5">IN</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>3</sub></font>(), <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font>
 <span class="kw">and </span><br/><a NAME="E13:7_1"/><span class="lab"><font color="Green" title="E10">A12</font></span>: 
<font color="Maroon" title="c5">IN</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c4">iN</font>
 <span class="kw">and </span><br/><a NAME="E14:7_1"/>
 for <font color="Olive" title="b1">K</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>3</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font></span>)</span>  st <font color="Olive" title="b1">K</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>3</sub></font>(), <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font> &amp; <font color="Olive" title="b1">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c4">iN</font> holds <br/><font color="Maroon" title="c5">IN</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>;<br/>
<a NAME="E15:7_1"/>
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font></span>)</span> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D9" onmouseover="rs('msualg_2/D9')" onmouseout="rh()">MSUALG_2:def 9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">h</font> =  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font></span>)</span> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font></span>)</span>,<font color="Maroon">F<sub>3</sub></font>() <span class="kw">by</span> <span class="lab"><a class="ref" href="extens_1.html#T5" onmouseover="rs('extens_1/T5')" onmouseout="rh()">EXTENS_1:5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">HIN</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()<span class="kw"> such that </span><br/><a NAME="E18:7_1"/>
<font color="Maroon" title="c7">HIN</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">and </span><br/><a NAME="E19:7_1"/>
<font color="Maroon" title="c7">HIN</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c4">iN</font>
 <span class="kw">and </span><br/><a NAME="E20:7_1"/><span class="lab"><font color="Green" title="E11">A13</font></span>: 
 for <font color="Olive" title="b1">K</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()  st <font color="Olive" title="b1">K</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive" title="b1">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c4">iN</font> holds <br/><font color="Maroon" title="c7">HIN</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E4">A2</font></span></a></span>;<br/>
<a NAME="E21:7_1"/>
<font color="Maroon" title="c6">h</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font>,<font color="Maroon">F<sub>3</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T22" onmouseover="rs('msualg_3/T22')" onmouseout="rh()">MSUALG_3:22</a></span>;<br/>
<a NAME="E22:7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A14</font></span>: 
<font color="Maroon" title="c6">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font>,<font color="Maroon">F<sub>3</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#D9" onmouseover="rs('msualg_3/D9')" onmouseout="rh()">MSUALG_3:def 9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">hIN</font> = <font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">IN</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() ;<br/>
<a NAME="E24:7_1"/>
<font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c4">iN</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">IN</font></span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>4</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:7_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="ref" href="pboole.html#T140" onmouseover="rs('pboole/T140')" onmouseout="rh()">PBOOLE:140</a></span>;<br/>
<a NAME="E25:7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A15</font></span>: 
<font color="Maroon" title="c7">HIN</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c8">hIN</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:7_1"><span class="lab"><font color="Green" title="E9">A11</font></span></a>, <a class="txt" href="#E20:7_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E22:7_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="ref" href="msualg_3.html#T10" onmouseover="rs('msualg_3/T10')" onmouseout="rh()">MSUALG_3:10</a></span>;<br/>
<a NAME="E26:7_1"/><span class="lab"><font color="Green" title="E14">A16</font></span>: 
<font color="Maroon">F<sub>3</sub></font>() is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#T5" onmouseover="rs('msualg_2/T5')" onmouseout="rh()">MSUALG_2:5</a></span>;<br/>
<a NAME="E27:7_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c4">iN</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T4" onmouseover="rs('msualg_3/T4')" onmouseout="rh()">MSUALG_3:4</a></span>;<br/>
<a NAME="E28:7_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c6">h</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c4">iN</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T4" onmouseover="rs('msualg_3/T4')" onmouseout="rh()">MSUALG_3:4</a></span>;<br/>
<a NAME="E29:7_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">HIN</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:7_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="ref" href="msualg_3.html#T9" onmouseover="rs('msualg_3/T9')" onmouseout="rh()">MSUALG_3:9</a></span>;<br/>
<a NAME="E30:7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A17</font></span>: 
<font color="Maroon" title="c7">HIN</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> 
 
;<br/>
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font></span>)</span> = 
<font color="Maroon" title="c6">h</font> <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="equation.html#T15" onmouseover="rs('equation/T15')" onmouseout="rh()">EQUATION:15</a></span>
<br/>.= 
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>()
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:7_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E30:7_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="ref" href="equation.html#T2" onmouseover="rs('equation/T2')" onmouseout="rh()">EQUATION:2</a>, <a class="ref" href="equation.html#T9" onmouseover="rs('equation/T9')" onmouseout="rh()">EQUATION:9</a></span>
;<br/>
<a NAME="E32:7_1"/><span class="kw">then </span>
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c2">Gen</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:7_1"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="ref" href="msualg_2.html#T9" onmouseover="rs('msualg_2/T9')" onmouseout="rh()">MSUALG_2:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E33:7_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K9" title="PBOOLE:func.9">.:.:</a> <font color="Maroon">F<sub>2</sub></font>() is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>()  <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon">F<sub>3</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#T3" onmouseover="rs('msafree/T3')" onmouseout="rh()">MSAFREE:3</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>
