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<span class="kw">let </span><font color="Maroon" title="c3">IT1</font>, <font color="Maroon" title="c4">IT2</font> be   <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</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"> ( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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">a</font>,<font color="Olive" title="b2">b</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="c3">IT1</font> iff (  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b2">b</font> or (  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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">o</font> ) ) ) ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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">a</font>,<font color="Olive" title="b2">b</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="c4">IT2</font> iff (  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b2">b</font> or (  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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">o</font> ) ) ) ) implies <font color="Maroon" title="c3">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">IT2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_1_2"/><span class="lab"><font color="Green" title="E27">A36</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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">a</font>,<font color="Olive" title="b2">b</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="c3">IT1</font> iff (  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b2">b</font> or (  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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">o</font> ) ) )
 <span class="kw">and </span><br/><a NAME="E2:42_1_2"/><span class="lab"><font color="Green" title="E28">A37</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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">a</font>,<font color="Olive" title="b2">b</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="c4">IT2</font> iff (  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b2">b</font> or (  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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">o</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="c3">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">IT2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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">a</font>,<font color="Maroon" title="c6">b</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="c3">IT1</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</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="c4">IT2</font> ) &amp; ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</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="c4">IT2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</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="c3">IT1</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</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="c4">IT2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</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="c3">IT1</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:42_1_2_1_1"/><span class="lab"><font color="Green" title="E29">A38</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</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="c3">IT1</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">a</font>,<font color="Maroon" title="c6">b</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="c4">IT2</font></span><br/><a NAME="E2:42_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A39</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">IT1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/><a NAME="E3:42_1_2_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">IT1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_2_1_1"><span class="lab"><font color="Green" title="E29">A38</font></span></a>, <a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a9</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b9</font> = <font color="Maroon" title="c6">b</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_2_1_1"><span class="lab"><font color="Green" title="E30">A39</font></span></a>, <a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E5:42_1_2_1_1"/>
(  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c7">a9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c8">b9</font> or (  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c8">b9</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c8">b9</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">o</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_2"><span class="lab"><font color="Green" title="E27">A36</font></span></a>, <a class="txt" href="#E1:42_1_2_1_1"><span class="lab"><font color="Green" title="E29">A38</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:42_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</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="c4">IT2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_2"><span class="lab"><font color="Green" title="E28">A37</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/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:42_1_2_1"/><span class="lab"><font color="Green" title="E29">A40</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</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="c4">IT2</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">a</font>,<font color="Maroon" title="c6">b</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="c3">IT1</font></span><br/><a NAME="E3:42_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A41</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">IT2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/><a NAME="E4:42_1_2_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">IT2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_2_1"><span class="lab"><font color="Green" title="E29">A40</font></span></a>, <a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a9</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b9</font> = <font color="Maroon" title="c6">b</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_2_1"><span class="lab"><font color="Green" title="E30">A41</font></span></a>, <a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E6:42_1_2_1"/>
(  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c7">a9</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c8">b9</font> or (  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c7">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bagorder.html#K3" title="BAGORDER:func.3">TotDegree</a> <font color="Maroon" title="c8">b9</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a9</font>,<font color="Maroon" title="c8">b9</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">o</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_2"><span class="lab"><font color="Green" title="E28">A37</font></span></a>, <a class="txt" href="#E2:42_1_2_1"><span class="lab"><font color="Green" title="E29">A40</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:42_1_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</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="c3">IT1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_2"><span class="lab"><font color="Green" title="E27">A36</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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:42_1_2"/>
<font color="Maroon" title="c3">IT1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">IT2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</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/>


</div>
