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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</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">  for <font color="Olive" title="b1">A</font> being    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon" title="c1">S</font>  st  ex <font color="Olive" title="b2">X</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> st  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">A</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Olive" title="b2">X</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> holds <br/><font color="Olive" title="b1">A</font>, <a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon" title="c1">S</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">X</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> st  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Olive" title="b1">X</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span> implies <font color="Maroon" title="c2">A</font>, <a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:36"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 ex <font color="Olive" title="b1">X</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> st  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Olive" title="b1">X</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</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"> <font color="Maroon" title="c2">A</font>, <a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">I</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">SB</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">ST</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">X</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E3:36"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c6">X</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E14">A1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">F</font> =  the    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">F1</font> =  the    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <span class="p1"><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">{</a><span class="default"><font color="Maroon" title="c6">X</font></span><a href="pzfmisc1.html#K1" title="PZFMISC1:func.1">}</a></span>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E15">A2</font></span></a></span>;<br/>
<span class="kw">take </span>
 the    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msualg_3.html#D11" onmouseover="rs('msualg_3/D11')" onmouseout="rh()">MSUALG_3:def 11</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">  the    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a> <font color="Maroon" title="c2">A</font>, <a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span><br/>

<a NAME="E5:36"/>
<font color="Maroon" title="c2">A</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36"><span class="lab"><font color="Green" title="E15">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:36"/>
 the    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a> <font color="Maroon" title="c2">A</font>, <a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_9.html#T25" target="_self" onmouseover="rs('msualg_9/T25')" onmouseout="rh()">Th25</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msualg_3.html#D10" onmouseover="rs('msualg_3/D10')" onmouseout="rh()">MSUALG_3:def 10</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">  the    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a> <font color="Maroon" title="c2">A</font>, <a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span><br/>

<span class="kw">hence </span><a NAME="E7:36"/>
 the    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">A</font>, <a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msualg_3.html#D9" onmouseover="rs('msualg_3/D9')" onmouseout="rh()">MSUALG_3:def 9</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">  the    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> </span><br/>

<a NAME="E8:36"/>
<font color="Maroon" title="c8">F1</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_9.html#T10" target="_self" onmouseover="rs('msualg_9/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:36"/>
 the    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree2.html#K7" title="MSAFREE2:func.7">Trivial_Algebra</a> <font color="Maroon" title="c1">S</font></span>)</span> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</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><br/>


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