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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">U0</font> being   <a href="msafree.html#V2" title="MSAFREE:attr.2">free</a>   <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b2">A</font> being   <a href="msafree.html#V1" title="MSAFREE:attr.1">free</a>   <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Olive" title="b1">U0</font><br/>  for <font color="Olive" title="b3">Z</font> being   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Olive" title="b1">U0</font>  st <font color="Olive" title="b3">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">A</font> &amp;  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Olive" title="b3">Z</font> is  <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  holds <br/> <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Olive" title="b3">Z</font> is  <a href="msafree.html#V2" title="MSAFREE:attr.2">free</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">U0</font> be   <a href="msafree.html#V2" title="MSAFREE:attr.2">free</a>   <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>   <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">  for <font color="Olive" title="b1">A</font> being   <a href="msafree.html#V1" title="MSAFREE:attr.1">free</a>   <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">U0</font><br/>  for <font color="Olive" title="b2">Z</font> being   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font>  st <font color="Olive" title="b2">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">A</font> &amp;  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Olive" title="b2">Z</font> is  <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  holds <br/> <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Olive" title="b2">Z</font> is  <a href="msafree.html#V2" title="MSAFREE:attr.2">free</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="msafree.html#V1" title="MSAFREE:attr.1">free</a>   <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon" title="c2">U0</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">Z</font> being   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font>  st <font color="Olive" title="b1">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Olive" title="b1">Z</font> is  <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  holds <br/> <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Olive" title="b1">Z</font> is  <a href="msafree.html#V2" title="MSAFREE:attr.2">free</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">Z</font> be   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</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">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font> is  <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</a>  implies  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font> is  <a href="msafree.html#V2" title="MSAFREE:attr.2">free</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c4">Z</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E2:34"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font> is  <a href="msualg_6.html#V1" title="MSUALG_6:attr.1">feasible</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">  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font> is  <a href="msafree.html#V2" title="MSAFREE:attr.2">free</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">Z1</font> = <font color="Maroon" title="c4">Z</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D17" onmouseover="rs('msualg_2/D17')" onmouseout="rh()">MSUALG_2:def 17</a></span>;<br/>
<a NAME="E4:34"/>
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c5">Z1</font></span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="extens_1.html#T18" onmouseover="rs('extens_1/T18')" onmouseout="rh()">EXTENS_1:18</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c4">Z</font> as    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D4" onmouseover="rs('msafree/D4')" onmouseout="rh()">MSAFREE:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">z1</font> = <font color="Maroon" title="c6">z</font> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c3">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="pboole.html#D18" onmouseover="rs('pboole/D18')" onmouseout="rh()">PBOOLE:def 18</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">z</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msafree.html#D6" onmouseover="rs('msafree/D6')" onmouseout="rh()">MSAFREE:def 6</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="c6">z</font> is  <a href="msafree.html#V1" title="MSAFREE:attr.1">free</a> </span><br/>

<a NAME="E7:34"/>
<font color="Maroon" title="c6">z</font> is  <a href="msafree.html#V1" title="MSAFREE:attr.1">free</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c8">D</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font></span>)</span> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D9" onmouseover="rs('msualg_2/D9')" onmouseout="rh()">MSUALG_2:def 9</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">U1</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" title="c1">S</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msafree.html#D5" onmouseover="rs('msafree/D5')" onmouseout="rh()">MSAFREE:def 5</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">b<sub>1</sub></font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c6">z</font>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U1</font>  ex <font color="Olive">b<sub>2</sub></font> being    <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 <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font></span>)</span>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U1</font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font>,<font color="Maroon" title="c9">U1</font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c6">z</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c10">g</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c6">z</font>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U1</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">b<sub>1</sub></font> being    <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 <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font></span>)</span>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U1</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font>,<font color="Maroon" title="c9">U1</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c6">z</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c10">g</font> )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c11">G</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">A</font>, the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c9">U1</font><span class="kw"> such that </span><br/><a NAME="E3:34_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c11">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">z1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c10">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="equation.html#T6" target="_self" onmouseover="rs('equation/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">h</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c9">U1</font><span class="kw"> such that </span><br/><a NAME="E5:34_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c9">U1</font>
 <span class="kw">and </span><br/><a NAME="E6:34_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c11">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D5" onmouseover="rs('msafree/D5')" onmouseout="rh()">MSAFREE:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">H</font> = <font color="Maroon" title="c12">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c8">D</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font></span>)</span>,<font color="Maroon" title="c9">U1</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">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="c13">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font>,<font color="Maroon" title="c9">U1</font> &amp; <font color="Maroon" title="c13">H</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c6">z</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c10">g</font> )</span><br/>

<span class="kw">thus </span><a NAME="E8:34_1"/>
<font color="Maroon" title="c13">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c4">Z</font>,<font color="Maroon" title="c9">U1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E5:34_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="equation.html#T22" target="_self" onmouseover="rs('equation/T22')" onmouseout="rh()">Th22</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="c13">H</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c6">z</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c10">g</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c10">g</font> = 
<font color="Maroon" title="c12">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c4">Z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E6:34_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="equation.html#T5" target="_self" onmouseover="rs('equation/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
<font color="Maroon" title="c13">H</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c6">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="equation.html#T5" target="_self" onmouseover="rs('equation/T5')" onmouseout="rh()">Th5</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">hence </span><a NAME="E8:34"/>
<font color="Maroon" title="c6">z</font> is  <a href="msafree.html#V1" title="MSAFREE:attr.1">free</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/>


</div>
