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<span class="kw">set </span><font color="Maroon" title="c2">S</font> =  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a> ;<br/>
<span class="kw">thus </span><a NAME="E1:29_1_1"/>
 <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="funcsdom.html#V1" title="FUNCSDOM:attr.1">strict</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">vector-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:29_1_1"/>
 <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">vector-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D2" onmouseover="rs('rlvect_1/D2')" onmouseout="rh()">RLVECT_1:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T11" target="_self" onmouseover="rs('asympt_3/T11')" onmouseout="rh()">LM12</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font> = 
<font color="Maroon" title="c5">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T14" target="_self" onmouseover="rs('asympt_3/T14')" onmouseout="rh()">LM16</a></span>
<br/>.= 
<font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T14" target="_self" onmouseover="rs('asympt_3/T14')" onmouseout="rh()">LM16</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">thus </span><a NAME="E3:29_1_1"/>
 <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">vector-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">z</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c8">z1</font> = <font color="Maroon" title="c5">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T11" target="_self" onmouseover="rs('asympt_3/T11')" onmouseout="rh()">LM12</a></span>;<br/>
<a NAME="E2:29_1_1_2"/><span class="lab"><font color="Green" title="E19">D1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T14" target="_self" onmouseover="rs('asympt_3/T14')" onmouseout="rh()">LM16</a></span>;<br/>
<a NAME="E3:29_1_1_2"/><span class="lab"><font color="Green" title="E20">D2</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T14" target="_self" onmouseover="rs('asympt_3/T14')" onmouseout="rh()">LM16</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">z1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_2"><span class="lab"><font color="Green" title="E19">D1</font></span></a>, <a class="ref" href="asympt_3.html#T14" target="_self" onmouseover="rs('asympt_3/T14')" onmouseout="rh()">LM16</a></span>
<br/>.= 
<font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">z1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_2"><span class="lab"><font color="Green" title="E20">D2</font></span></a>, <a class="ref" href="asympt_3.html#T14" target="_self" onmouseover="rs('asympt_3/T14')" onmouseout="rh()">LM16</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">thus </span><a NAME="E4:29_1_1"/>
 <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">vector-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T11" target="_self" onmouseover="rs('asympt_3/T11')" onmouseout="rh()">LM12</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a></span>)</span> = 
<font color="Maroon" title="c4">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T16" target="_self" onmouseover="rs('asympt_3/T16')" onmouseout="rh()">LM18</a>, <a class="ref" href="asympt_3.html#T14" target="_self" onmouseover="rs('asympt_3/T14')" onmouseout="rh()">LM16</a></span>
<br/>.= 
<font color="Maroon" title="c3">x</font>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">thus </span><a NAME="E5:29_1_1"/>
 <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">vector-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D16" onmouseover="rs('algstr_0/D16')" onmouseout="rh()">ALGSTR_0:def 16</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T11" target="_self" onmouseover="rs('asympt_3/T11')" onmouseout="rh()">LM12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">t1</font> = <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font>;<br/>
<span class="kw">take </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D11" onmouseover="rs('algstr_0/D11')" onmouseout="rh()">ALGSTR_0:def 11</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a></span><br/>

<a NAME="E2:29_1_1_4"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E3:29_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">D1</font></span>: 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T15" target="_self" onmouseover="rs('asympt_3/T15')" onmouseout="rh()">LM17</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span> = 
<font color="Maroon" title="c4">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_4"><span class="lab"><font color="Green" title="E19">D1</font></span></a>, <a class="ref" href="asympt_3.html#T14" target="_self" onmouseover="rs('asympt_3/T14')" onmouseout="rh()">LM16</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#T11" onmouseover="rs('funcsdom/T11')" onmouseout="rh()">FUNCSDOM:11</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#D3" target="_self" onmouseover="rs('asympt_3/D3')" onmouseout="rh()">defAlgebra</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>

<a NAME="E6:29_1_1"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a> holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T11" target="_self" onmouseover="rs('asympt_3/T11')" onmouseout="rh()">LM12</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font> = 
<font color="Maroon" title="c5">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">y1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T13" target="_self" onmouseover="rs('asympt_3/T13')" onmouseout="rh()">LM15</a></span>
<br/>.= 
<font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T13" target="_self" onmouseover="rs('asympt_3/T13')" onmouseout="rh()">LM15</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E7:29_1_1"/>
 <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">vector-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  )</span><br/>

<a NAME="E8:29_1_1"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c8">z1</font> = <font color="Maroon" title="c5">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T11" target="_self" onmouseover="rs('asympt_3/T11')" onmouseout="rh()">LM12</a></span>;<br/>
<a NAME="E2:29_1_1_6"/><span class="lab"><font color="Green" title="E19">D1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T13" target="_self" onmouseover="rs('asympt_3/T13')" onmouseout="rh()">LM15</a></span>;<br/>
<a NAME="E3:29_1_1_6"/><span class="lab"><font color="Green" title="E20">D2</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T13" target="_self" onmouseover="rs('asympt_3/T13')" onmouseout="rh()">LM15</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">y1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">z1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_6"><span class="lab"><font color="Green" title="E19">D1</font></span></a>, <a class="ref" href="asympt_3.html#T13" target="_self" onmouseover="rs('asympt_3/T13')" onmouseout="rh()">LM15</a></span>
<br/>.= 
<font color="Maroon" title="c6">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">z1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_6"><span class="lab"><font color="Green" title="E20">D2</font></span></a>, <a class="ref" href="asympt_3.html#T13" target="_self" onmouseover="rs('asympt_3/T13')" onmouseout="rh()">LM15</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E9:29_1_1"/>
 <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">vector-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E10:29_1_1"/>
 <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">vector-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_1.html#D4" onmouseover="rs('vectsp_1/D4')" onmouseout="rh()">VECTSP_1:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c3">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T11" target="_self" onmouseover="rs('asympt_3/T11')" onmouseout="rh()">LM12</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a></span>)</span> = 
<font color="Maroon" title="c4">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T17" target="_self" onmouseover="rs('asympt_3/T17')" onmouseout="rh()">LM19</a>, <a class="ref" href="asympt_3.html#T13" target="_self" onmouseover="rs('asympt_3/T13')" onmouseout="rh()">LM15</a></span>
<br/>.= 
<font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D4" onmouseover="rs('vectsp_1/D4')" onmouseout="rh()">VECTSP_1:def 4</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>

<a NAME="E11:29_1_1"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a> holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</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="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c8">z1</font> = <font color="Maroon" title="c5">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T11" target="_self" onmouseover="rs('asympt_3/T11')" onmouseout="rh()">LM12</a></span>;<br/>
<a NAME="E2:29_1_1_8"/><span class="lab"><font color="Green" title="E19">D1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T13" target="_self" onmouseover="rs('asympt_3/T13')" onmouseout="rh()">LM15</a></span>;<br/>
<a NAME="E3:29_1_1_8"/><span class="lab"><font color="Green" title="E20">D2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T13" target="_self" onmouseover="rs('asympt_3/T13')" onmouseout="rh()">LM15</a></span>;<br/>
<a NAME="E4:29_1_1_8"/><span class="lab"><font color="Green" title="E21">D3</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T14" target="_self" onmouseover="rs('asympt_3/T14')" onmouseout="rh()">LM16</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">z</font></span>)</span> = 
<font color="Maroon" title="c6">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">z1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1_1_8"><span class="lab"><font color="Green" title="E21">D3</font></span></a>, <a class="ref" href="asympt_3.html#T13" target="_self" onmouseover="rs('asympt_3/T13')" onmouseout="rh()">LM15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">y1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">z1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D2" onmouseover="rs('vectsp_1/D2')" onmouseout="rh()">VECTSP_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</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="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_8"><span class="lab"><font color="Green" title="E19">D1</font></span></a>, <a class="txt" href="#E3:29_1_1_8"><span class="lab"><font color="Green" title="E20">D2</font></span></a>, <a class="ref" href="asympt_3.html#T14" target="_self" onmouseover="rs('asympt_3/T14')" onmouseout="rh()">LM16</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E12:29_1_1"/>
 <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">vector-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  )</span><br/>

<a NAME="E13:29_1_1"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a><br/>  for <font color="Olive" title="b3">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Olive" title="b3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Olive" title="b1">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">s</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T11" target="_self" onmouseover="rs('asympt_3/T11')" onmouseout="rh()">LM12</a></span>;<br/>
<a NAME="E2:29_1_1_9"/><span class="lab"><font color="Green" title="E19">D1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T13" target="_self" onmouseover="rs('asympt_3/T13')" onmouseout="rh()">LM15</a></span>;<br/>
<a NAME="E3:29_1_1_9"/><span class="lab"><font color="Green" title="E20">D3</font></span>: 
<font color="Maroon" title="c5">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E4:29_1_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">D2</font></span>: 
<font color="Maroon" title="c5">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T15" target="_self" onmouseover="rs('asympt_3/T15')" onmouseout="rh()">LM17</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c5">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font></span>)</span> = 
<font color="Maroon" title="c5">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T15" target="_self" onmouseover="rs('asympt_3/T15')" onmouseout="rh()">LM17</a>, <a class="txt" href="#E2:29_1_1_9"><span class="lab"><font color="Green" title="E19">D1</font></span></a>, <a class="txt" href="#E3:29_1_1_9"><span class="lab"><font color="Green" title="E20">D3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">y1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcsdom.html#D9" onmouseover="rs('funcsdom/D9')" onmouseout="rh()">FUNCSDOM:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T13" target="_self" onmouseover="rs('asympt_3/T13')" onmouseout="rh()">LM15</a>, <a class="txt" href="#E4:29_1_1_9"><span class="lab"><font color="Green" title="E21">D2</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E14:29_1_1"/>
 <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">vector-associative</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  )</span><br/>

<a NAME="E15:29_1_1"/>
 for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">x</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">t</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T11" target="_self" onmouseover="rs('asympt_3/T11')" onmouseout="rh()">LM12</a></span>;<br/>
<a NAME="E2:29_1_1_10"/><span class="lab"><font color="Green" title="E19">D0</font></span>: 
( <font color="Maroon" title="c3">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c4">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E3:29_1_1_10"/><span class="lab"><font color="Green" title="E20">D2</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E4:29_1_1_10"/><span class="lab"><font color="Green" title="E21">D1</font></span>: 
<font color="Maroon" title="c4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_10"><span class="lab"><font color="Green" title="E19">D0</font></span></a>, <a class="ref" href="asympt_3.html#T15" target="_self" onmouseover="rs('asympt_3/T15')" onmouseout="rh()">LM17</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1_1_10"><span class="lab"><font color="Green" title="E20">D2</font></span></a>, <a class="ref" href="asympt_3.html#T15" target="_self" onmouseover="rs('asympt_3/T15')" onmouseout="rh()">LM17</a></span>
<br/>.= 
<font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" onmouseover="rs('rlvect_1/D7')" onmouseout="rh()">RLVECT_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29_1_1_10"><span class="lab"><font color="Green" title="E19">D0</font></span></a>, <a class="txt" href="#E4:29_1_1_10"><span class="lab"><font color="Green" title="E21">D1</font></span></a>, <a class="ref" href="asympt_3.html#T15" target="_self" onmouseover="rs('asympt_3/T15')" onmouseout="rh()">LM17</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E16:29_1_1"/>
 <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  )</span><br/>

<a NAME="E17:29_1_1"/>
 for <font color="Olive" title="b1">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a> holds  <font color="Olive" title="b1">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">y</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="29_1_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">s</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a> holds  <font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</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="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</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="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T11" target="_self" onmouseover="rs('asympt_3/T11')" onmouseout="rh()">LM12</a></span>;<br/>
<a NAME="E2:29_1_1_11"/><span class="lab"><font color="Green" title="E19">D1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T14" target="_self" onmouseover="rs('asympt_3/T14')" onmouseout="rh()">LM16</a></span>;<br/>
<a NAME="E3:29_1_1_11"/><span class="lab"><font color="Green" title="E20">D3</font></span>: 
<font color="Maroon" title="c3">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E4:29_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">D2</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T15" target="_self" onmouseover="rs('asympt_3/T15')" onmouseout="rh()">LM17</a></span>;<br/>
<a NAME="E5:29_1_1_11"/><span class="lab"><font color="Green" title="E22">D4</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T15" target="_self" onmouseover="rs('asympt_3/T15')" onmouseout="rh()">LM17</a>, <a class="txt" href="#E3:29_1_1_11"><span class="lab"><font color="Green" title="E20">D3</font></span></a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> = 
<font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T15" target="_self" onmouseover="rs('asympt_3/T15')" onmouseout="rh()">LM17</a>, <a class="txt" href="#E2:29_1_1_11"><span class="lab"><font color="Green" title="E19">D1</font></span></a>, <a class="txt" href="#E3:29_1_1_11"><span class="lab"><font color="Green" title="E20">D3</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D5" onmouseover="rs('rlvect_1/D5')" onmouseout="rh()">RLVECT_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</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="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T14" target="_self" onmouseover="rs('asympt_3/T14')" onmouseout="rh()">LM16</a>, <a class="txt" href="#E4:29_1_1_11"><span class="lab"><font color="Green" title="E21">D2</font></span></a>, <a class="txt" href="#E5:29_1_1_11"><span class="lab"><font color="Green" title="E22">D4</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E18:29_1_1"/>
 <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>  is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">t</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_1.html#D6" onmouseover="rs('rlvect_1/D6')" onmouseout="rh()">RLVECT_1:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>1</sub></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="c4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</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="c4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c7">t</font> = <font color="Maroon" title="c4">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c5">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" title="FUNCSDOM:func.12">RAlgebra</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T11" target="_self" onmouseover="rs('asympt_3/T11')" onmouseout="rh()">LM12</a></span>;<br/>
<a NAME="E21:29_1_1"/><span class="lab"><font color="Green" title="E19">D2</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T15" target="_self" onmouseover="rs('asympt_3/T15')" onmouseout="rh()">LM17</a></span>;<br/>
<a NAME="E22:29_1_1"/><span class="lab"><font color="Green" title="E20">D4</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T15" target="_self" onmouseover="rs('asympt_3/T15')" onmouseout="rh()">LM17</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="asympt_3.html#T15" target="_self" onmouseover="rs('asympt_3/T15')" onmouseout="rh()">LM17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" onmouseover="rs('rlvect_1/D6')" onmouseout="rh()">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</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="c7">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:29_1_1"><span class="lab"><font color="Green" title="E19">D2</font></span></a>, <a class="txt" href="#E22:29_1_1"><span class="lab"><font color="Green" title="E20">D4</font></span></a>, <a class="ref" href="asympt_3.html#T14" target="_self" onmouseover="rs('asympt_3/T14')" onmouseout="rh()">LM16</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E24:29_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</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="c4">t</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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