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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b1</font>, <font color="Olive" title="b2">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b2">b2</font> &amp; <font color="Olive" title="b2">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Olive" title="b1">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b2">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">b1</font></span><br/><b>let </b><font color="Maroon" title="c2">b1</font>, <font color="Maroon" title="c3">b2</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b2</font> &amp; <font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c2">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c3">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">b1</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:110"/><i><font color="Green" title="E61">A1</font></i>: 
<font color="Maroon" title="c2">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c3">b2</font>
 <b>and </b><br/><a NAME="E2:110"/><i><font color="Green" title="E62">A2</font></i>: 
<font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c2">b1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">b1</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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 <font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_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="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font> )</span><br/><b>assume </b><a NAME="E1:110_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_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="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span><br/><a NAME="E2:110_1"/><i><font color="Green" title="E63">A3</font></i>: 
<font color="Maroon" title="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="polynom1.html#D13" target="_self" title="POLYNOM1:def.13">Def13</a></i>;<br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">-'</a> <font color="Maroon" title="c2">b1</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
<b>by </b><i><a class="txt" href="#E2:110"><i><font color="Green" title="E62">A2</font></i></a>, <a class="ref" href="polynom1.html#T56" target="_self" title="POLYNOM1:th.56">Th56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_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="p1">(<span class="default"><font color="Maroon" title="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D6" target="_self" title="POLYNOM1:def.6">Def6</a></i>
;<br/><a NAME="E4:110_1"/><b>then </b>
<font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></i>;<br/><b>hence </b><a NAME="E5:110_1"/>
<font color="Maroon" title="c3">b2</font> <a href="recdef_1.html#K1" title="RECDEF_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="c2">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="txt" href="#E2:110_1"><i><font color="Green" title="E63">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:110"/>
<font color="Maroon" title="c3">b2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">b1</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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