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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">L</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">r</font> being  <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Olive" title="b3">q</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Olive" title="b4">r</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b2">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">q</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Olive" title="b4">r</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</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">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being  <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Olive" title="b3">r</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Olive" title="b3">r</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">r</font> be   <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c5">r</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c5">r</font></span>)</span></span><br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="147_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">b</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21" target="_self" title="POLYNOM1:def.21">Def21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21" target="_self" title="POLYNOM1:def.21">Def21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21" target="_self" title="POLYNOM1:def.21">Def21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c5">r</font></span>)</span></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="ref" href="polynom1.html#D21" target="_self" title="POLYNOM1:def.21">Def21</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="E2:147"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c5">r</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K20" title="POLYNOM1:func.20">+</a> <font color="Maroon" title="c5">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</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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