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<a NAME="E1:13_1"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
 for <font color="Olive" title="b1">I</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">F</font> being    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">MSAlgebra-Family</a> of <font color="Olive" title="b1">I</font>,<font color="Maroon">F<sub>1</sub></font>()  st (  for <font color="Olive" title="b3">i</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b3">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">I</font> holds <br/> ex <font color="Olive" title="b4">A</font> being    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive" title="b4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b4">A</font>] ) ) holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Olive" title="b2">F</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:13"><span class="lab"><font color="Green" title="E8">A6</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c1">V</font> =  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#NK1" title="NUMBERS:NK.1">NAT</a>;<br/>
<a NAME="E2:13_1"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">B</font> being   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Olive" title="b1">A</font>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">B</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>;<br/>
<a NAME="E3:13_1"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
 for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">B</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">A</font> being   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c3">fi</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#NK1" title="NUMBERS:NK.1">NAT</a>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E5:13_1"/><span class="lab"><font color="Green" title="E12">A10</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">A</font>]
 <span class="kw">and </span><br/><a NAME="E6:13_1"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
 for <font color="Olive" title="b1">B</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">G</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#NK1" title="NUMBERS:NK.1">NAT</a>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">B</font>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/> ex <font color="Olive" title="b3">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">B</font> st <br/>( <font color="Olive" title="b3">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b3">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">fi</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">G</font> &amp; (  for <font color="Olive" title="b4">K</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">B</font>  st <font color="Olive" title="b4">K</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b4">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">fi</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b3">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b4">K</font> ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="birkhoff.html#S2" target="_self" onmouseover="rs('birkhoff/S2')" onmouseout="rh()">BIRKHOFF:sch 2</a>(<a class="txt" href="#E3:13_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="txt" href="#E2:13_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>);<br/></span>
<a NAME="E7:13_1"/><span class="lab"><font color="Green" title="E14">A12</font></span>: 
 for <font color="Olive" title="b1">C</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">G</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#NK1" title="NUMBERS:NK.1">NAT</a>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">C</font>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">C</font>] holds <br/> ex <font color="Olive" title="b3">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">C</font> st <br/>( <font color="Olive" title="b3">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b3">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">G</font> &amp; (  for <font color="Olive" title="b4">K</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">C</font>  st <font color="Olive" title="b4">K</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b4">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b3">H</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b4">K</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E4">A2</font></span></a></span>;<br/>
<a NAME="E8:13_1"/><span class="lab"><font color="Green" title="E15">A13</font></span>: 
 for <font color="Olive" title="b1">C</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">G</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#NK1" title="NUMBERS:NK.1">NAT</a>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">C</font>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">C</font>] holds <br/> ex <font color="Olive" title="b3">h</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">C</font> st <br/>( <font color="Olive" title="b3">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b2">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b3">h</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <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="13_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">C</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <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">  for <font color="Olive" title="b1">G</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#NK1" title="NUMBERS:NK.1">NAT</a>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">C</font>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">C</font>] holds <br/> ex <font color="Olive" title="b2">h</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c4">C</font> st <br/>( <font color="Olive" title="b2">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c4">C</font> &amp; <font color="Olive" title="b1">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">h</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>3</sub></font>() )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">G</font> be    <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#NK1" title="NUMBERS:NK.1">NAT</a>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">C</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"> ( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">C</font>] implies  ex <font color="Olive" title="b1">h</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c4">C</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">h</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>3</sub></font>() ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:13_1_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">C</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">  ex <font color="Olive" title="b1">h</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c4">C</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">h</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>3</sub></font>() )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">h</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c4">C</font><span class="kw"> such that </span><br/><a NAME="E3:13_1_1"/><span class="lab"><font color="Green" title="E16">A14</font></span>: 
<font color="Maroon" title="c6">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E4:13_1_1"/><span class="lab"><font color="Green" title="E17">A15</font></span>: 
<font color="Maroon" title="c5">G</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">F<sub>3</sub></font>()
 <span class="kw">and </span><br/><a NAME="E5:13_1_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>2</sub></font>(),<font color="Maroon" title="c4">C</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>2</sub></font>(),<font color="Maroon" title="c4">C</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>3</sub></font>() <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c5">G</font> holds <br/><font color="Maroon" title="c6">h</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="#E2:13"><span class="lab"><font color="Green" title="E4">A2</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">h</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"> ( <font color="Maroon" title="c6">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">G</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">F<sub>3</sub></font>() )</span><br/>

<span class="kw">thus </span><a NAME="E6:13_1_1"/>
( <font color="Maroon" title="c6">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c5">G</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">F<sub>3</sub></font>() )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13_1_1"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="txt" href="#E4:13_1_1"><span class="lab"><font color="Green" title="E17">A15</font></span></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="E9:13_1"/><span class="lab"><font color="Green" title="E16">A16</font></span>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>;<br/>
<a NAME="E10:13_1"/><span class="lab"><font color="Green" title="E17">A17</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">A</font>] iff  for <font color="Olive" title="b2">s</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b3">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s</font>  st (  for <font color="Olive" title="b4">B</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b4">B</font>] holds <br/><font color="Olive" title="b4">B</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b3">e</font> ) holds <br/><font color="Olive" title="b1">A</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b3">e</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E3">A1</font></span></a></span>;<br/>
<a NAME="E11:13_1"/><span class="lab"><font color="Green" title="E18">A18</font></span>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c2">A</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="birkhoff.html#S10" target="_self" onmouseover="rs('birkhoff/S10')" onmouseout="rh()">BIRKHOFF:sch 10</a>(<a class="txt" href="#E10:13_1"><span class="lab"><font color="Green" title="E17">A17</font></span></a>, <a class="txt" href="#E5:13_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>);<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c4">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c2">A</font><span class="kw"> such that </span><br/><a NAME="E13:13_1"/><span class="lab"><font color="Green" title="E19">A19</font></span>: 
<font color="Maroon" title="c4">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E14:13_1"/><span class="lab"><font color="Green" title="E20">A20</font></span>: 
<font color="Maroon" title="c3">fi</font> <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a>  <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c4">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a></span>)</span>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="birkhoff.html#S3" target="_self" onmouseover="rs('birkhoff/S3')" onmouseout="rh()">BIRKHOFF:sch 3</a>(<a class="txt" href="#E11:13_1"><span class="lab"><font color="Green" title="E18">A18</font></span></a>, <a class="txt" href="#E8:13_1"><span class="lab"><font color="Green" title="E15">A13</font></span></a>);<br/></span>
<a NAME="E15:13_1"/><span class="lab"><font color="Green" title="E21">A21</font></span>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a>  <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <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#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>,<font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="birkhoff.html#D1" target="_self" onmouseover="rs('birkhoff/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_1_2"><span class="kw">now </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">  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI: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>() holds <br/><font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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"> ( <font color="Maroon" title="c5">i</font> <a href="tarski.html#R2" title="TARSKI: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>() implies <font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:13_1_2"/>
<font color="Maroon" title="c5">i</font> <a href="tarski.html#R2" title="TARSKI: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"> <font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c5">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">thus </span><a NAME="E3:13_1_2"/>
<font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_1_2_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:13_1_2_1"/><span class="lab"><font color="Green" title="E22">A22</font></span>: 
 for <font color="Olive" title="b1">D</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">E</font> being   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Olive" title="b1">D</font>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">D</font>] holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b2">E</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">D</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <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">  for <font color="Olive" title="b1">E</font> being   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c7">D</font>  st <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c7">D</font>] holds <br/><font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">E</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c8">E</font> be   <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a>   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c7">D</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"> ( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c7">D</font>] implies <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c8">E</font>] )</span><br/>



<span class="kw">assume </span><a NAME="E1:13_1_2_1_1"/><span class="lab"><font color="Green" title="E23">A23</font></span>: 
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c7">D</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"> <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c8">E</font>]</span><br/>

<a NAME="E2:13_1_2_1_1"/>
 for <font color="Olive" title="b1">s</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">e</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" title="EQUATION:func.4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>  st (  for <font color="Olive" title="b3">B</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b3">B</font>] holds <br/><font color="Olive" title="b3">B</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b2">e</font> ) holds <br/><font color="Maroon" title="c8">E</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Olive" title="b2">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E1:13_1_2_1_1"><span class="lab"><font color="Green" title="E23">A23</font></span></a>, <a class="ref" href="equation.html#T31" onmouseover="rs('equation/T31')" onmouseout="rh()">EQUATION:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:13_1_2_1_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c8">E</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E3">A1</font></span></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="E2:13_1_2_1"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a>  <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a>  <a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <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#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>,<font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="birkhoff.html#S6" target="_self" onmouseover="rs('birkhoff/S6')" onmouseout="rh()">BIRKHOFF:sch 6</a>(<a class="txt" href="#E7:13_1"><span class="lab"><font color="Green" title="E14">A12</font></span></a>, <a class="txt" href="#E9:13_1"><span class="lab"><font color="Green" title="E16">A16</font></span></a>, <a class="txt" href="#E1:13_1_2_1"><span class="lab"><font color="Green" title="E22">A22</font></span></a>);<br/></span>
<a NAME="E3:13_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A24</font></span>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a>  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_3.html#D8" onmouseover="rs('msualg_3/D8')" onmouseout="rh()">MSUALG_3:def 8</a></span>;<br/>
<a NAME="E4:13_1_2_1"/><span class="lab"><font color="Green" title="E24">A25</font></span>: 
 for <font color="Olive" title="b1">C</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">G</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#NK1" title="NUMBERS:NK.1">NAT</a>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">C</font>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">C</font>] holds <br/> ex <font color="Olive" title="b3">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">C</font> st <br/>( <font color="Olive" title="b3">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">C</font> &amp; <font color="Olive" title="b2">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b3">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">fi</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_1_2_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">C</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <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">  for <font color="Olive" title="b1">G</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#NK1" title="NUMBERS:NK.1">NAT</a>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">C</font>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">C</font>] holds <br/> ex <font color="Olive" title="b2">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">C</font> st <br/>( <font color="Olive" title="b2">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">C</font> &amp; <font color="Olive" title="b1">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">fi</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">G</font> be    <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#NK1" title="NUMBERS:NK.1">NAT</a>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">C</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"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">C</font>] implies  ex <font color="Olive" title="b1">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">C</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c8">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">fi</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:13_1_2_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">C</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">  ex <font color="Olive" title="b1">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">C</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c8">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b1">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">fi</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">C</font><span class="kw"> such that </span><br/><a NAME="E3:13_1_2_1_2"/><span class="lab"><font color="Green" title="E25">A26</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">C</font>
 <span class="kw">and </span><br/><a NAME="E4:13_1_2_1_2"/><span class="lab"><font color="Green" title="E26">A27</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">fi</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c8">G</font>
 <span class="kw">and </span><br/><a NAME="E5:13_1_2_1_2"/>
 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" title="c2">A</font>,<font color="Maroon" title="c7">C</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" title="c2">A</font>,<font color="Maroon" title="c7">C</font> &amp; <font color="Olive" title="b1">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">fi</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c8">G</font> holds <br/><font color="Maroon" title="c9">H</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="#E6:13_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">H</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"> ( <font color="Maroon" title="c9">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c8">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c9">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">fi</font> )</span><br/>

<span class="kw">thus </span><a NAME="E6:13_1_2_1_2"/>
( <font color="Maroon" title="c9">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">C</font> &amp; <font color="Maroon" title="c8">G</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c9">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c3">fi</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13_1_2_1_2"><span class="lab"><font color="Green" title="E25">A26</font></span></a>, <a class="txt" href="#E4:13_1_2_1_2"><span class="lab"><font color="Green" title="E26">A27</font></span></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>
<span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</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="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</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="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> or  not <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</font> or <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:13_1_2_1"/><span class="lab"><font color="Green" title="E25">A28</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:13_1_2_1"/><span class="lab"><font color="Green" title="E26">A29</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:13_1_2_1"/><span class="lab"><font color="Green" title="E27">A30</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b</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"> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font></span><br/>

<span class="lab"><font color="Green" title="E28">A31</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> = 
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13_1_2_1"><span class="lab"><font color="Green" title="E23">A24</font></span></a>, <a class="ref" href="msualg_3.html#D3" onmouseover="rs('msualg_3/D3')" onmouseout="rh()">MSUALG_3:def 3</a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">t1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E10:13_1_2_1"/><span class="lab"><font color="Green" title="E29">A32</font></span>: 
<font color="Maroon" title="c9">t1</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"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:13_1_2_1"/><span class="lab"><font color="Green" title="E30">A33</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_1_2_1"><span class="lab"><font color="Green" title="E25">A28</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">t2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E13:13_1_2_1"/><span class="lab"><font color="Green" title="E31">A34</font></span>: 
<font color="Maroon" title="c10">t2</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"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:13_1_2_1"/><span class="lab"><font color="Green" title="E32">A35</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:13_1_2_1"><span class="lab"><font color="Green" title="E26">A29</font></span></a>, <a class="txt" href="#E8:13_1_2_1"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">t1</font> = <font color="Maroon" title="c9">t1</font>, <font color="Maroon" title="c12">t2</font> = <font color="Maroon" title="c10">t2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="equation.html#K3" title="EQUATION:func.3">TermAlg</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:13_1_2_1"><span class="lab"><font color="Green" title="E29">A32</font></span></a>, <a class="txt" href="#E13:13_1_2_1"><span class="lab"><font color="Green" title="E31">A34</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">e</font> = <font color="Maroon" title="c11">t1</font> <a href="equation.html#K5" title="EQUATION:func.5">'='</a> <font color="Maroon" title="c12">t2</font>;<br/>
<span class="lab"><font color="Green" title="E33">A36</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">fi</font> <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">t1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">t1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:13_1"><span class="lab"><font color="Green" title="E20">A20</font></span></a>, <a class="ref" href="msualg_3.html#T2" onmouseover="rs('msualg_3/T2')" onmouseout="rh()">MSUALG_3:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:13_1_2_1"><span class="lab"><font color="Green" title="E30">A33</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">t2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:13_1_2_1"><span class="lab"><font color="Green" title="E27">A30</font></span></a>, <a class="txt" href="#E14:13_1_2_1"><span class="lab"><font color="Green" title="E32">A35</font></span></a>, <a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">fi</font> <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">t2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:13_1"><span class="lab"><font color="Green" title="E20">A20</font></span></a>, <a class="ref" href="msualg_3.html#T2" onmouseover="rs('msualg_3/T2')" onmouseout="rh()">MSUALG_3:2</a></span>
;<br/>
<a NAME="E17:13_1_2_1"/>
 for <font color="Olive" title="b1">B</font> being   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Olive" title="b1">B</font> <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c11">t1</font> <a href="equation.html#K5" title="EQUATION:func.5">'='</a> <font color="Maroon" title="c12">t2</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="birkhoff.html#S4" target="_self" onmouseover="rs('birkhoff/S4')" onmouseout="rh()">BIRKHOFF:sch 4</a>(<a class="txt" href="#E16:13_1_2_1"><span class="lab"><font color="Green" title="E33">A36</font></span></a>, <a class="txt" href="#E4:13_1_2_1"><span class="lab"><font color="Green" title="E24">A25</font></span></a>);<br/></span>
<a NAME="E18:13_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A37</font></span>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="equation.html#R1" title="EQUATION:pred.1">|=</a> <font color="Maroon" title="c11">t1</font> <a href="equation.html#K5" title="EQUATION:func.5">'='</a> <font color="Maroon" title="c12">t2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c7">a</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">t1</font> <a href="equation.html#K5" title="EQUATION:func.5">'='</a> <font color="Maroon" title="c12">t2</font></span>)</span> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:13_1_2_1"><span class="lab"><font color="Green" title="E30">A33</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">t1</font> <a href="equation.html#K5" title="EQUATION:func.5">'='</a> <font color="Maroon" title="c12">t2</font></span>)</span> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:13_1"><span class="lab"><font color="Green" title="E21">A21</font></span></a>, <a class="txt" href="#E18:13_1_2_1"><span class="lab"><font color="Green" title="E34">A37</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:13_1_2_1"><span class="lab"><font color="Green" title="E32">A35</font></span></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></div><span class="kw">end;</span></div>
<a NAME="E17:13_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">H</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T1" onmouseover="rs('msualg_3/T1')" onmouseout="rh()">MSUALG_3:1</a></span>;<br/>
<a NAME="E18:13_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">H</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:13_1"><span class="lab"><font color="Green" title="E19">A19</font></span></a>, <a class="ref" href="msualg_3.html#D9" onmouseover="rs('msualg_3/D9')" onmouseout="rh()">MSUALG_3:def 9</a></span>;<br/>
<a NAME="E19:13_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A38</font></span>: 
<font color="Maroon">F<sub>2</sub></font>(), <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c4">H</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_9.html#T16" onmouseover="rs('msualg_9/T16')" onmouseout="rh()">MSUALG_9:16</a></span>;<br/>
<a NAME="E20:13_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_3.html#K6" title="MSUALG_3:func.6">Image</a> <font color="Maroon" title="c4">H</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E5:13_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:13_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="#E4:13"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E19:13_1"><span class="lab"><font color="Green" title="E22">A38</font></span></a>, <a class="ref" href="msualg_3.html#T17" onmouseover="rs('msualg_3/T17')" onmouseout="rh()">MSUALG_3:17</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>
