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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">H</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 );<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E2:45_1_1"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X</font> iff ( <font color="Olive" title="b1">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Y</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xfamily.html#S1" onmouseover="rs('xfamily/S1')" onmouseout="rh()">XFAMILY:sch 1</a>();<br/></span>
<span class="kw">set </span><font color="Maroon" title="c4">F</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">A1</font> =  <a href="funct_6.html#K3" title="FUNCT_6:func.3">meet</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">X</font></span>)</span>;<br/>
<a NAME="E3:45_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">X</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is   <a href="algstr_4.html#V2" title="ALGSTR_4:attr.2">stable</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</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="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">X</font></span>)</span> implies <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> is   <a href="algstr_4.html#V2" title="ALGSTR_4:attr.2">stable</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_1_1_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">X</font></span>)</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"> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> is   <a href="algstr_4.html#V2" title="ALGSTR_4:attr.2">stable</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font></span><br/>

<a NAME="E2:45_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:45_1_1_1"/>
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> is   <a href="algstr_4.html#V2" title="ALGSTR_4:attr.2">stable</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">A1</font> =  <a href="funct_6.html#K3" title="FUNCT_6:func.3">meet</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">X</font></span>)</span> as   <a href="algstr_4.html#V2" title="ALGSTR_4:attr.2">stable</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T9" target="_self" onmouseover="rs('algstr_4/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">N1</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #);<br/>
<span class="kw">take </span>
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</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"> (  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) &amp; (  for <font color="Olive" title="b1">N</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font>  st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N</font> holds <br/> <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is    <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Olive" title="b1">N</font> ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:45_1_1_2"/>
( <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:45_1_1_2_1"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" 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="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) &amp; (  for <font color="Olive" title="b1">N</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font>  st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N</font> holds <br/> <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is    <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Olive" title="b1">N</font> ) )</span><br/><div class="add"><a NAME="E2:45_1_1_2_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D1" onmouseover="rs('zfmisc_1/D1')" onmouseout="rh()">ZFMISC_1:def 1</a></span>;<br/><a NAME="E3:45_1_1_2_1"/>
 ex <font color="Olive" title="b1">H</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">H</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:45_1_1_2_1_1"/>
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">M</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">M</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<a NAME="E2:45_1_1_2_1_1"/><span class="kw">then </span>
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">M</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="realset1.html#D2" onmouseover="rs('realset1/D2')" onmouseout="rh()">REALSET1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">H</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">M</font> #) as   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D9" target="_self" onmouseover="rs('algstr_4/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">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"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">H</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> )</span><br/>

<span class="kw">thus </span><a NAME="E4:45_1_1_2_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">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="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span><br/>

<span class="kw">thus </span><a NAME="E5:45_1_1_2_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:45_1_1_2_1"/>
(  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) &amp; (  for <font color="Olive" title="b1">N</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font>  st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N</font> holds <br/> <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is    <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Olive" title="b1">N</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1_2_1"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E2:45_1_1_2_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E19">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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:45_1_1_2_2"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.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"> (  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) &amp; (  for <font color="Olive" title="b1">N</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font>  st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N</font> holds <br/> <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is    <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Olive" title="b1">N</font> ) )</span><br/><div class="add"><a NAME="E2:45_1_1_2_2"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_1_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_1_1_2_2_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A1</font></span><br/>

<a NAME="E2:45_1_1_2_2_1"/>
 for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_1_2_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">Y</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="c9">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_1_1_2_2_1_1"/>
<font color="Maroon" title="c9">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X</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="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">H</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font><span class="kw"> such that </span><br/><a NAME="E3:45_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
( <font color="Maroon" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:45_1_1_2_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E1:45_1_1_2_2_1"><span class="lab"><font color="Green" title="E22">A7</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="E3:45_1_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_1_1_2_2"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 1</a></span>;<br/>
<a NAME="E4:45_1_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:45_1_1_2_2_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#D4" onmouseover="rs('funct_6/D4')" onmouseout="rh()">FUNCT_6:def 4</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="E3:45_1_1_2_2"/>
 for <font color="Olive" title="b1">N</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font>  st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N</font> holds <br/> <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is    <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Olive" title="b1">N</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_1_2_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">N</font> be   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</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="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</font> implies  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is    <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c8">N</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_1_1_2_2_2"/><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</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">  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is    <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c8">N</font></span><br/>

<a NAME="E2:45_1_1_2_2_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="45_1_1_2_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_1_1_2_2_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</font></span><br/>

<a NAME="E2:45_1_1_2_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">X</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#D4" onmouseover="rs('funct_6/D4')" onmouseout="rh()">FUNCT_6:def 4</a></span>;<br/>
<a NAME="E3:45_1_1_2_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c3">X</font>
 
;<br/>
<a NAME="E4:45_1_1_2_2_2_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D9" target="_self" onmouseover="rs('algstr_4/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E5:45_1_1_2_2_2_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:45_1_1_2_2_2"><span class="lab"><font color="Green" title="E22">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:45_1_1_2_2_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1_2_2_2_1"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:45_1_1_2_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A11</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</font>
 
;<br/>
<span class="lab"><font color="Green" title="E24">A12</font></span>:  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">M</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">M</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="realset1.html#D2" onmouseover="rs('realset1/D2')" onmouseout="rh()">REALSET1:def 2</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c8">N</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D9" target="_self" onmouseover="rs('algstr_4/D9')" onmouseout="rh()">Def9</a></span>
;<br/>
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) = 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">M</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #), the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="realset1.html#D2" onmouseover="rs('realset1/D2')" onmouseout="rh()">REALSET1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">M</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">N</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #), the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1_2_2_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="relat_1.html#T74" onmouseover="rs('relat_1/T74')" onmouseout="rh()">RELAT_1:74</a>, <a class="ref" href="zfmisc_1.html#T96" onmouseover="rs('zfmisc_1/T96')" onmouseout="rh()">ZFMISC_1:96</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c8">N</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1_1_2_2_2"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="realset1.html#D2" onmouseover="rs('realset1/D2')" onmouseout="rh()">REALSET1:def 2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:45_1_1_2_2_2"/>
 <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is    <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c8">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1_1_2_2_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="algstr_4.html#D9" target="_self" onmouseover="rs('algstr_4/D9')" onmouseout="rh()">Def9</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="E4:45_1_1_2_2"/>
(  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) &amp; (  for <font color="Olive" title="b1">N</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font>  st <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N</font> holds <br/> <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c6">A1</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c6">A1</font></span>)</span> #) is    <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Olive" title="b1">N</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1_1_2_2"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="algstr_4.html#D9" target="_self" onmouseover="rs('algstr_4/D9')" onmouseout="rh()">Def9</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>

</div>
