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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">b1</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Olive" title="b2">b2</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <font color="Olive" title="b1">b1</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Olive" title="b1">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">b2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">b1</font>, <font color="Maroon" title="c3">b2</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c3">b2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c2">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">b2</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:113"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
<font color="Maroon" title="c2">b1</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c3">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c2">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">b2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">k</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:113_1"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">k9</font> = <font color="Maroon" title="c4">k</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><a NAME="E3:113_1"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:113"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="pre_poly.html#D11" target="_self" title="PRE_POLY:def.11">Def11</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D5" target="_self" title="PRE_POLY:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D6" target="_self" title="PRE_POLY:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:113_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="nat_d.html#T38" title="NAT_D:th.38">NAT_D:38</a></span>
<br/>.= 
<font color="Maroon" title="c3">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
; <a class="txt" 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="E3:113"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c2">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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