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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b1">x9</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Olive" title="b2">y9</font> );<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">IR</font> being   <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:53_1_1"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IR</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="relset_1.html#S1" title="RELSET_1:sch.1">RELSET_1:sch 1</a>();<br/></span>
<span class="kw">set </span><font color="Maroon" title="c3">IT</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">IR</font> #);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">IT</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">IR</font> #) as   <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">IT</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c4">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">IT</font> iff <font color="Olive" title="b1">x</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Olive" title="b2">y</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:53_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">IT</font> iff <font color="Olive" title="b1">x</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Olive" title="b2">y</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">IT</font> iff <font color="Maroon" title="c5">x</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c6">y</font> )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c6">y</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">IT</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:53_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">IT</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c6">y</font></span><br/><a NAME="E2:53_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x9</font>, <font color="Olive" title="b2">y9</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Olive" title="b2">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Olive" title="b1">x9</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Olive" title="b2">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:53_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c6">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E6:53_1_1"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c6">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">IT</font></span><br/>

<a NAME="E7:53_1_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/>
<a NAME="E8:53_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:53_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">IT</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1_1"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E6:53_1_1"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E7:53_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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