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<span class="kw">let </span><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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">f1</font>, <font color="Olive" title="b2">g1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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>  st <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> &amp; <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">g1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">g1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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>; <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="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> &amp; <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g1</font> implies <font color="Maroon" title="c2">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">g1</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:26"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
( <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> &amp; <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g1</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"> <font color="Maroon" title="c2">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">g1</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">X</font> =  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">S</font> =  <a href="asympt_3.html#K2" title="ASYMPT_3:func.2">R_Algebra_of_Big_Oh_poly</a> ;<br/>
<a NAME="E2:26"/><span class="lab"><font color="Green" title="E16">A3a</font></span>: 
 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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">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>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c2">f</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">g</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6" title="FUNCSDOM:func.6">RealFuncMult</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="asympt_3.html#K1" title="ASYMPT_3:func.1">Big_Oh_poly</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<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>
<br/>.= 
<font color="Maroon" title="c4">f1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c5">g1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E16">A3a</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</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/>


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