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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a><br/>  for <font color="Olive" title="b2">v1</font> being   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v1</font> holds <br/><span class="p1">(<span class="default"><a href="zmodul01.html#K2" title="ZMODUL01:func.2">Int-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">v1</font></span><br/><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="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</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">v1</font> being   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</a>  st <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v1</font> holds <br/><span class="p1">(<span class="default"><a href="zmodul01.html#K2" title="ZMODUL01:func.2">Int-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">v1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">v1</font> be   <a href="rat_1.html#NM1" title="RAT_1:NM.1">Rational</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v1</font> implies <span class="p1">(<span class="default"><a href="zmodul01.html#K2" title="ZMODUL01:func.2">Int-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">v1</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:31"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v1</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="zmodul01.html#K2" title="ZMODUL01:func.2">Int-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">v1</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">xx</font> = <font color="Maroon" title="c1">x</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" onmouseover="rs('int_1/D2')" onmouseout="rh()">INT_1:def 2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:31_1"/>
( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1"><span class="kw">suppose </span></a><a NAME="E1:31_1_1"/><span class="lab"><font color="Green" title="E18">C1</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"><a href="zmodul01.html#K2" title="ZMODUL01:func.2">Int-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">v1</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x0</font> = <font color="Maroon" title="c1">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="zmodul01.html#K2" title="ZMODUL01:func.2">Int-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">v</font>) = 
<span class="p1">(<span class="default"><a href="binom.html#K3" title="BINOM:func.3">Nat-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c4">xx</font>,<font color="Maroon" title="c2">v</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1"><span class="lab"><font color="Green" title="E18">C1</font></span></a>, <a class="ref" href="zmodul01.html#D20" onmouseover="rs('zmodul01/D20')" onmouseout="rh()">ZMODUL01:def 20</a></span>
<br/>.= 
<font color="Maroon" title="c5">x0</font> <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="zmodul07.html#T14" target="_self" onmouseover="rs('zmodul07/T14')" onmouseout="rh()">LMTFRat1</a>, <a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E17">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">v1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2"><span class="kw">suppose </span></a><a NAME="E1:31_1_2"/><span class="lab"><font color="Green" title="E18">C2</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"><a href="zmodul01.html#K2" title="ZMODUL01:func.2">Int-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">v1</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">x0</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="zmodul01.html#K2" title="ZMODUL01:func.2">Int-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">v</font>) = 
<span class="p1">(<span class="default"><a href="binom.html#K3" title="BINOM:func.3">Nat-mult-left</a> <a href="gaussint.html#K11" title="GAUSSINT:func.11">F_Rat</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c4">xx</font></span>)</span>,<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">v</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_2"><span class="lab"><font color="Green" title="E18">C2</font></span></a>, <a class="ref" href="zmodul01.html#D20" onmouseover="rs('zmodul01/D20')" onmouseout="rh()">ZMODUL01:def 20</a></span>
<br/>.= 
<font color="Maroon" title="c5">x0</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="txt" href="#E1:31"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="zmodul07.html#T14" target="_self" onmouseover="rs('zmodul07/T14')" onmouseout="rh()">LMTFRat1</a></span>
<br/>.= 
<font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">v1</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></div><span class="kw">end;</span></div>

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