<?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/>
<span class="kw">set </span><font color="Maroon" title="c2">sA</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>();<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">Gen</font> being    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon">F<sub>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E2:9_1"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<font color="Maroon" title="c3">Gen</font> is  <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree2.html#D10" title="MSAFREE2:def.10">MSAFREE2:def 10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">Gen</font> = <font color="Maroon" title="c3">Gen</font> as  <a href="finset_1.html#V2" title="FINSET_1:attr.2">V42</a>()  <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>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">GEN</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V8</a>() <a href="finset_1.html#V2" title="FINSET_1:attr.2">V42</a>()  <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>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E5:9_1"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
<font color="Maroon" title="c4">Gen</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c5">GEN</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_9.html#T7" title="MSUALG_9:th.7">MSUALG_9:7</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">F</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<font color="Maroon" title="c5">GEN</font><span class="kw"> such that </span><br/><a NAME="E7:9_1"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<font color="Maroon" title="c6">F</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_9.html#T12" title="MSUALG_9:th.12">MSUALG_9:12</a></span>;<br/>
<a NAME="E8:9_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> <font color="Maroon" title="c5">GEN</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pzfmisc1.html#D3" title="PZFMISC1:def.3">PZFMISC1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">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 <font color="Maroon" title="c5">GEN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_1_1"/>
<font color="Maroon" title="c7">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">GEN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">hence </span><a NAME="E2:9_1_1"/>
(  not <font color="Maroon" title="c5">GEN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" 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="c7">G</font> = <font color="Maroon" title="c6">F</font> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="ref" href="extens_1.html#T5" title="EXTENS_1:th.5">EXTENS_1:5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">h</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>2</sub></font>()<span class="kw"> such that </span><br/><a NAME="E11:9_1"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<font color="Maroon" title="c8">h</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>2</sub></font>()
 <span class="kw">and </span><br/><a NAME="E12:9_1"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">h</font> <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="#E1:9"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">sI1</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_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c8">h</font></span>)</span> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c8">h</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<a NAME="E14:9_1"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
<font color="Maroon" title="c5">GEN</font> is    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="msscyc_1.html#T21" title="MSSCYC_1:th.21">MSSCYC_1:21</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">sI</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_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c8">h</font></span>)</span> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D9" title="MSUALG_2:def.9">MSUALG_2:def 9</a></span>;<br/>
<a NAME="E16:9_1"/>
<font color="Maroon" title="c5">GEN</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c10">sI</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a>,<a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">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 <font color="Maroon" title="c5">GEN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">sI</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:9_1_2"/>
<font color="Maroon" title="c11">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">GEN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">sI</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">s</font> = <font color="Maroon" title="c11">i</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<span class="kw">let </span><font color="Maroon" title="c13">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">GEN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> or <font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">sI</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:9_1_2"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">GEN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">sI</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font></span><br/>

<a NAME="E4:9_1_2"/><span class="lab"><font color="Green" title="E15">A13</font></span>: 
<span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">fi</font> = <font color="Maroon">F<sub>4</sub></font>() <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<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="c12">s</font></span>)</span> ;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">fi</font></span>)</span> = 
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">fi</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/>
<a NAME="E7:9_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A14</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">fi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T15" title="FUNCT_1:th.15">FUNCT_1:15</a></span>;<br/>
<a NAME="E8:9_1_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">GEN</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:9_1_2"/><span class="lab"><font color="Green" title="E17">A15</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:9_1_2"/><span class="lab"><font color="Green" title="E18">A16</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_2"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> = 
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9_1_2"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">fi</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/>
<a NAME="E13:9_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A17</font></span>: 
<font color="Maroon" title="c14">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">fi</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:9_1_2"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<font color="Maroon" title="c13">g</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">fi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:9_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="txt" href="#E11:9_1_2"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:9_1_2"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
;<br/>
<a NAME="E15:9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <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="c12">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:9_1_2"><span class="lab"><font color="Green" title="E19">A17</font></span></a>, <a class="txt" href="#E7:9_1_2"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E16:9_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">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 <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D20" title="PBOOLE:def.20">PBOOLE:def 20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:9_1_2"/>
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">sI</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:9_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="msualg_3.html#D12" title="MSUALG_3:def.12">MSUALG_3:def 12</a></span>; <a class="txt" 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="E17:9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A18</font></span>: 
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c5">GEN</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c10">sI</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="extens_1.html#T17" title="EXTENS_1:th.17">EXTENS_1:17</a></span>;<br/>
<a NAME="E18:9_1"/>
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c10">sI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c9">sI1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="extens_1.html#T18" title="EXTENS_1:th.18">EXTENS_1:18</a></span>;<br/>
<a NAME="E19:9_1"/><span class="kw">then </span>
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c5">GEN</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c8">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:9_1"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="ref" href="msualg_2.html#T21" title="MSUALG_2:th.21">MSUALG_2:21</a></span>;<br/>
<a NAME="E20:9_1"/><span class="kw">then </span>
<font color="Maroon">F<sub>2</sub></font>() is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c8">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:9_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="msafree.html#T3" title="MSAFREE:th.3">MSAFREE:3</a></span>;<br/>
<a NAME="E21:9_1"/><span class="kw">then </span>
<font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c8">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#T7" title="MSUALG_2:th.7">MSUALG_2:7</a></span>;<br/>
<a NAME="E22:9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A19</font></span>: 
<font color="Maroon" title="c8">h</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:9_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="ref" href="msualg_3.html#T19" title="MSUALG_3:th.19">MSUALG_3:19</a></span>;<br/>
<a NAME="E23:9_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K13" title="MSUALG_4:func.13">QuotMSAlg</a> (<font color="Maroon">F<sub>3</sub></font>(),<span class="p1">(<span class="default"><a href="msualg_4.html#K17" title="MSUALG_4:func.17">MSCng</a> <font color="Maroon" title="c8">h</font></span>)</span>)]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:9_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E22:9_1"><span class="lab"><font color="Green" title="E15">A19</font></span></a>, <a class="ref" href="msualg_4.html#T6" title="MSUALG_4:th.6">MSUALG_4:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
