<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c1">ZS</font> =  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #);<br/>
<a NAME="E1:31_1_1"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds  <font color="Olive" title="b1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c3">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</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="c2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">v1</font> = <font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c5">w1</font> = <font color="Maroon" title="c3">w</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">w</font> = 
<font color="Maroon" title="c4">v1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">w1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</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="E2:31_1_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</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="c2">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">u1</font> = <font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c7">w1</font> = <font color="Maroon" title="c4">w</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E2:31_1_1_2"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c2">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E3:31_1_1_2"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">w1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">u1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">w1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</a>, <a class="txt" href="#E2:31_1_1_2"><span class="lab"><font color="Green" title="E15">A3</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">u1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">w1</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</a>, <a class="txt" href="#E3:31_1_1_2"><span class="lab"><font color="Green" title="E16">A4</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>
<a NAME="E3:31_1_1"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
 for <font color="Olive" title="b1">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds  <font color="Olive" title="b1">v</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="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</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="c2">v</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="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">v1</font> = <font color="Maroon" title="c2">v</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c2">v</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="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #)</span>)</span> = 
<font color="Maroon" title="c3">v1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<font color="Maroon" title="c2">v</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>
<a NAME="E4:31_1_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
 for <font color="Olive" title="b1">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds  <font color="Olive" title="b1">v</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</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="c2">v</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<span class="kw">take </span>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c2">v</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="c2">v</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c2">v</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="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">v1</font> = <font color="Maroon" title="c2">v</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c2">v</font> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">v1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="gaussint.html#E14"><span class="lab"><font color="Green" title="E9">LmX</font></span></a>, <a class="ref" href="gaussint.html#T5" target="_self" onmouseover="rs('gaussint/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">v1</font>

;<br/>
<span class="kw">hence </span><font color="Maroon" title="c2">v</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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a></span>)</span></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c2">v</font></span>)</span> = 
<font color="Maroon" title="c3">v1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">v1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #)

;<br/> <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="E5:31_1_1"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a><br/>  for <font color="Olive" title="b3">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b3">v</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="b2">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</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">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">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="c3">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">v</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</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="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">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="c3">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c5">v1</font> = <font color="Maroon" title="c4">v</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E2:31_1_1_5"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
( <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font> &amp; <font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T5" target="_self" onmouseover="rs('gaussint/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T5" target="_self" onmouseover="rs('gaussint/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">v1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">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="c3">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1_1_5"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</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="E6:31_1_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a><br/>  for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds  <font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b2">v</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">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b3">w</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</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">v</font>, <font color="Olive" title="b2">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds  <font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">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="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b2">w</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</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="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c3">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="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c5">v1</font> = <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c6">w1</font> = <font color="Maroon" title="c4">w</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E2:31_1_1_6"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">w1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E3:31_1_1_6"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
( <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T5" target="_self" onmouseover="rs('gaussint/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span> = 
<font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">w1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1_1_6"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="gaussint.html#T5" target="_self" onmouseover="rs('gaussint/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">w1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c3">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="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_1_6"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</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="E7:31_1_1"/><span class="lab"><font color="Green" title="E19">A13</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a><br/>  for <font color="Olive" title="b3">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</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">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">v</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</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="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c5">v1</font> = <font color="Maroon" title="c4">v</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E2:31_1_1_7"/><span class="lab"><font color="Green" title="E20">A14</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T5" target="_self" onmouseover="rs('gaussint/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T5" target="_self" onmouseover="rs('gaussint/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">v1</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c4">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1_1_7"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="ref" href="gaussint.html#T5" target="_self" onmouseover="rs('gaussint/T5')" onmouseout="rh()">Th5</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="E8:31_1_1"/><span class="lab"><font color="Green" title="E20">A15</font></span>: 
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #)<br/>  for <font color="Olive" title="b3">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> holds  <font color="Olive" title="b3">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">w</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c3">w</font> be   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</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">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a> holds  <font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</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="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c5">v1</font> = <font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c6">w1</font> = <font color="Maroon" title="c3">w</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E2:31_1_1_8"/><span class="lab"><font color="Green" title="E21">A16</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T5" target="_self" onmouseover="rs('gaussint/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">vw</font> = <font color="Maroon" title="c2">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font></span>)</span> = 
<font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">vw</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T5" target="_self" onmouseover="rs('gaussint/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">w1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">w1</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">*</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a>, <a class="txt" href="#E2:31_1_1_8"><span class="lab"><font color="Green" title="E21">A16</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>
<a NAME="E9:31_1_1"/><span class="lab"><font color="Green" title="E21">A17</font></span>: 
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds  <font color="Olive" title="b1">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c3">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</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="c2">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">v1</font> = <font color="Maroon" title="c2">v</font>, <font color="Maroon" title="c5">w1</font> = <font color="Maroon" title="c3">w</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c2">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">w</font> = 
<font color="Maroon" title="c4">v1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">w1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>
<br/>.= 
<font color="Maroon" title="c3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</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="E10:31_1_1"/><span class="lab"><font color="Green" title="E22">A18</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">v</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="b2">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</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="c2">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">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="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">a1</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c6">b1</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c4">v</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">ab</font> = <font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">b</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E3:31_1_1_10"/><span class="lab"><font color="Green" title="E23">A19</font></span>: 
( <font color="Maroon" title="c5">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font> = 
<font color="Maroon" title="c8">ab</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">v1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">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="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_1_10"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</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="E11:31_1_1"/><span class="lab"><font color="Green" title="E23">A20</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds <br/> ( <font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">v</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">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">w</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</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="b3">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</font></span>)</span> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</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="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">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="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</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">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">a1</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c7">w1</font> = <font color="Maroon" title="c4">w</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">vw</font> = <font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E3:31_1_1_11"/><span class="lab"><font color="Green" title="E24">A21</font></span>: 
( <font color="Maroon" title="c5">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">v</font> &amp; <font color="Maroon" title="c5">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span> = 
<font color="Maroon" title="c5">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">vw</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>
<br/>.= 
<font color="Maroon" title="c5">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">w1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">w1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">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="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_1_11"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="ref" href="gaussint.html#T4" target="_self" onmouseover="rs('gaussint/T4')" onmouseout="rh()">Th4</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</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">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E5:31_1_1_11"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</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">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31_1_1"><span class="lab"><font color="Green" title="E22">A18</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>
<a NAME="E12:31_1_1"/><span class="lab"><font color="Green" title="E24">A22</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</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="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">a1</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c6">b1</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c4">v</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">ab</font> = <font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c9">bv</font> = <font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font> = 
<font color="Maroon" title="c8">ab</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>
<br/>.= 
<font color="Maroon" title="c5">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">v1</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c5">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">bv</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</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="E13:31_1_1"/>
 for <font color="Olive" title="b1">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) holds <br/> ( <font color="Olive" title="b1">v</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="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #)</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="31_1_1_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</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="c2">v</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="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #)</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">v1</font> = <font color="Maroon" title="c2">v</font> as   <a href="gaussint.html#NM1" title="GAUSSINT:NM.1">G_INTEG</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T2" target="_self" onmouseover="rs('gaussint/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c2">v</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="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #)</span>)</span> = 
<font color="Maroon" title="c3">v1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>
<br/>.= 
<font color="Maroon" title="c2">v</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"> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #)</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #)</span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">v</font> = 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T6" target="_self" onmouseover="rs('gaussint/T6')" onmouseout="rh()">Th6</a></span>
<br/>.= 
<font color="Maroon" title="c2">v</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">hence </span><a NAME="E14:31_1_1"/>
(  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) is  <a href="polyalg1.html#V1" title="POLYALG1:attr.1">strict</a>  &amp;  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  &amp;  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp;  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) is  <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a>  &amp;  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) is  <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a>  &amp;  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) is  <a href="polyalg1.html#V2" title="POLYALG1:attr.2">mix-associative</a>  &amp;  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) is  <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">scalar-associative</a>  &amp;  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) is  <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">vector-distributive</a>  &amp;  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>,<a href="gaussint.html#K2" title="GAUSSINT:func.2">g_int_add</a>,<a href="gaussint.html#K3" title="GAUSSINT:func.3">g_int_mult</a>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">In</a> (1,<a href="gaussint.html#K1" title="GAUSSINT:func.1">G_INT_SET</a>)</span>)</span>,<a href="gaussint.html#K4" title="GAUSSINT:func.4">Sc_Mult</a> #) is  <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">scalar-distributive</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E2:31_1_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E3:31_1_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E4:31_1_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E5:31_1_1"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E6:31_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a>, <a class="ref" href="rlvect_1.html#D2" onmouseover="rs('rlvect_1/D2')" onmouseout="rh()">RLVECT_1:def 2</a>, <a class="ref" href="rlvect_1.html#D3" onmouseover="rs('rlvect_1/D3')" onmouseout="rh()">RLVECT_1:def 3</a>, <a class="ref" href="rlvect_1.html#D4" onmouseover="rs('rlvect_1/D4')" onmouseout="rh()">RLVECT_1:def 4</a>, <a class="txt" href="#E9:31_1_1"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="txt" href="#E12:31_1_1"><span class="lab"><font color="Green" title="E24">A22</font></span></a>, <a class="txt" href="#E11:31_1_1"><span class="lab"><font color="Green" title="E23">A20</font></span></a>, <a class="txt" href="#E8:31_1_1"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="ref" href="group_1.html#D12" onmouseover="rs('group_1/D12')" onmouseout="rh()">GROUP_1:def 12</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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