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<span class="kw">set </span><font color="Maroon" title="c3">ZS</font> =  <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> ;<br/>
<a NAME="E1:85"/><span class="lab"><font color="Green" title="E35">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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</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="85_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_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">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c7">w1</font> = <font color="Maroon" title="c5">w</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> = 
<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#T27" target="_self" onmouseover="rs('gaussint/T27')" onmouseout="rh()">Th27</a></span>
<br/>.= 
<font color="Maroon" title="c5">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T27" target="_self" onmouseover="rs('gaussint/T27')" onmouseout="rh()">Th27</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="E2:85"/><span class="lab"><font color="Green" title="E36">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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</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="85_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_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"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">u1</font> = <font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c9">w1</font> = <font color="Maroon" title="c6">w</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E2:85_2"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c4">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T27" target="_self" onmouseover="rs('gaussint/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<a NAME="E3:85_2"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">w1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T27" target="_self" onmouseover="rs('gaussint/T27')" onmouseout="rh()">Th27</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">u1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">v1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">w1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T27" target="_self" onmouseover="rs('gaussint/T27')" onmouseout="rh()">Th27</a>, <a class="txt" href="#E2:85_2"><span class="lab"><font color="Green" title="E37">A3</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">u1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">w1</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c4">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T27" target="_self" onmouseover="rs('gaussint/T27')" onmouseout="rh()">Th27</a>, <a class="txt" href="#E3:85_2"><span class="lab"><font color="Green" title="E38">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="E3:85"/><span class="lab"><font color="Green" title="E37">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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</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="85_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_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">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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font></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#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c4">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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a></span>)</span> = 
<font color="Maroon" title="c5">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#T27" target="_self" onmouseover="rs('gaussint/T27')" onmouseout="rh()">Th27</a></span>
<br/>.= 
<font color="Maroon" title="c4">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="E4:85"/><span class="lab"><font color="Green" title="E38">A6</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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</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="85_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_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">v</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">v1</font> = <font color="Maroon" title="c4">v</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">w1</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T11" target="_self" onmouseover="rs('gaussint/T11')" onmouseout="rh()">Th11</a></span>;<br/>
<a NAME="E3:85_4"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<font color="Maroon" title="c6">w1</font> <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"><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="c5">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">w1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">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="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">w1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">w1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font></span>)</span> = 
<font color="Maroon" title="c5">v1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">v1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85_4"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="gaussint.html#T27" target="_self" onmouseover="rs('gaussint/T27')" onmouseout="rh()">Th27</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:85"/><span class="lab"><font color="Green" title="E39">A8</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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</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="85_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_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"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">v</font> <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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">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="c5">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">v</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">a1</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c8">b1</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c9">v1</font> = <font color="Maroon" title="c6">v</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">ab</font> = <font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">b</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E3:85_5"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
( <font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">v</font> &amp; <font color="Maroon" title="c8">b1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">v</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">v</font> = 
<font color="Maroon" title="c10">ab</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T27" target="_self" onmouseover="rs('gaussint/T27')" onmouseout="rh()">Th27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">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="c8">b1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">v1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">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="c5">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85_5"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="gaussint.html#T27" target="_self" onmouseover="rs('gaussint/T27')" onmouseout="rh()">Th27</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="E6:85"/><span class="lab"><font color="Green" title="E40">A10</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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</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="85_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">w</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">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="c6">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">a1</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c9">w1</font> = <font color="Maroon" title="c6">w</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">vw</font> = <font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E3:85_6"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
( <font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">v</font> &amp; <font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span> = 
<font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c10">vw</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</a></span>
<br/>.= 
<font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">w1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T27" target="_self" onmouseover="rs('gaussint/T27')" onmouseout="rh()">Th27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">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="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">w1</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:85_6"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="gaussint.html#T27" target="_self" onmouseover="rs('gaussint/T27')" onmouseout="rh()">Th27</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="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">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="c6">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E5:85_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">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="c6">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:85"><span class="lab"><font color="Green" title="E39">A8</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="E7:85"/><span class="lab"><font color="Green" title="E41">A12</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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> holds  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="85_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">a1</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">b1</font> = <font color="Maroon" title="c5">b</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font> = 
<font color="Maroon" title="c6">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c7">b1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</a></span>
<br/>.= 
<font color="Maroon" title="c5">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</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="E8:85"/><span class="lab"><font color="Green" title="E42">A13</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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</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="85_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_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"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">v</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">a1</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c8">b1</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c9">v1</font> = <font color="Maroon" title="c6">v</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">ab</font> = <font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c11">bv</font> = <font color="Maroon" title="c5">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">v</font> as   <a href="gaussint.html#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">v</font> = 
<font color="Maroon" title="c10">ab</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">v1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</a></span>
<br/>.= 
<font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">v1</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c7">a1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c11">bv</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</a></span>
<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</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="E9:85"/><span class="lab"><font color="Green" title="E43">A14</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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</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="85_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_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">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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a></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="c4">v</font> )</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#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c4">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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a></span>)</span> = 
<font color="Maroon" title="c5">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#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</a></span>
<br/>.= 
<font color="Maroon" title="c4">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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a></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="c4">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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font> = 
1 <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#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</a></span>
<br/>.= 
<font color="Maroon" title="c4">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="E10:85"/>
 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="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> holds <br/><font color="Olive" title="b1">v</font> is  <a href="algstr_0.html#V23" title="ALGSTR_0:attr.23">left_invertible</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="85_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_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">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> implies <font color="Maroon" title="c4">v</font> is  <a href="algstr_0.html#V23" title="ALGSTR_0:attr.23">left_invertible</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:85_10"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_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">v</font> is  <a href="algstr_0.html#V23" title="ALGSTR_0:attr.23">left_invertible</a> </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#NM3" title="GAUSSINT:NM.3">G_RAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T10" target="_self" onmouseover="rs('gaussint/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">w</font> = <font color="Maroon" title="c5">v1</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="gaussint.html#T11" target="_self" onmouseover="rs('gaussint/T11')" onmouseout="rh()">Th11</a></span>;<br/>
<font color="Maroon" title="c6">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">v1</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></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#T29" target="_self" onmouseover="rs('gaussint/T29')" onmouseout="rh()">Th29</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85_10"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" onmouseover="rs('xcmplx_0/D7')" onmouseout="rh()">XCMPLX_0:def 7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:85_10"/>
<font color="Maroon" title="c4">v</font> is  <a href="algstr_0.html#V23" title="ALGSTR_0:attr.23">left_invertible</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:85"/>
 <a href="gaussint.html#K14" title="GAUSSINT:func.14">Gauss_RAT_Ring</a>  is   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:85"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E2:85"><span class="lab"><font color="Green" title="E36">A2</font></span></a>, <a class="txt" href="#E3:85"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E4:85"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E6:85"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E8:85"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="txt" href="#E9:85"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E7:85"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="vectsp_1.html#D6" onmouseover="rs('vectsp_1/D6')" onmouseout="rh()">VECTSP_1:def 6</a>, <a class="ref" href="vectsp_1.html#D7" onmouseover="rs('vectsp_1/D7')" onmouseout="rh()">VECTSP_1:def 7</a>, <a class="ref" href="algstr_0.html#D39" onmouseover="rs('algstr_0/D39')" onmouseout="rh()">ALGSTR_0:def 39</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="group_1.html#D12" onmouseover="rs('group_1/D12')" onmouseout="rh()">GROUP_1:def 12</a>, <a class="ref" href="struct_0.html#D8" onmouseover="rs('struct_0/D8')" onmouseout="rh()">STRUCT_0:def 8</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="ref" href="algstr_0.html#D16" onmouseover="rs('algstr_0/D16')" onmouseout="rh()">ALGSTR_0:def 16</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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