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<b>let </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span></span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1"/>
( ( <font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) or <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  or <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  or <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  or <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>suppose </b></a><a NAME="E1:7_1_1"/>
( <font color="Maroon" title="c1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">f1</font> = <font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c4">g1</font> = <font color="Maroon" title="c2">g</font> as  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/><b>consider </b><font color="Maroon" title="c5">a</font> being  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E4:7_1_1"/><i><font color="Green" title="E4">A1</font></i>: 
( <font color="Maroon" title="c3">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  )
 <b>by </b><i><a class="ref" href="xxreal_3.html#D4" title="XXREAL_3:def.4">XXREAL_3:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">b</font> being  <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E6:7_1_1"/><i><font color="Green" title="E5">A2</font></i>: 
( <font color="Maroon" title="c4">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a>  )
 <b>by </b><i><a class="ref" href="xxreal_3.html#D4" title="XXREAL_3:def.4">XXREAL_3:def 4</a></i>;<br/><a NAME="E7:7_1_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being <a href="xcmplx_0.html#V1" title="XCMPLX_0:attr.1">complex</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  st <br/>( <font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">b</font> )
 <b>by </b><i><a class="txt" href="#E4:7_1_1"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="#E6:7_1_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></i>;<br/><a NAME="E8:7_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c4">g1</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> 
 <b>by </b><i><a class="txt" href="#E4:7_1_1"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="#E6:7_1_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="xcmplx_1.html#T205" title="XCMPLX_1:th.205">XCMPLX_1:205</a></i>;<br/><b>hence </b><a NAME="E9:7_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2"><b>suppose </b></a><a NAME="E1:7_1_2"/><i><font color="Green" title="E4">A3</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span></span><br/><div class="add"><a NAME="E2:7_1_2"/>
( <font color="Maroon" title="c2">g</font> is <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> or <font color="Maroon" title="c2">g</font> is <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> or <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E3:7_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  )
 <b>by </b><i><a class="txt" href="#E1:7_1_2"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  = 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E1:7_1_2"><i><font color="Green" title="E4">A3</font></i></a>, <a class="ref" href="xxreal_3.html#T1" title="XXREAL_3:th.1">XXREAL_3:1</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_3"><b>suppose </b></a><a NAME="E1:7_1_3"/><i><font color="Green" title="E4">A4</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span></span><br/><div class="add"><a NAME="E2:7_1_3"/>
( <font color="Maroon" title="c2">g</font> is <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> or <font color="Maroon" title="c2">g</font> is <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> or <font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E3:7_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  )
 <b>by </b><i><a class="txt" href="#E1:7_1_3"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  = 
<span class="p1">(<span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E1:7_1_3"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="xxreal_3.html#T1" title="XXREAL_3:th.1">XXREAL_3:1</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_4"><b>suppose </b></a><a NAME="E1:7_1_4"/><i><font color="Green" title="E4">A5</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span></span><br/><div class="add"><a NAME="E2:7_1_4"/>
( <font color="Maroon" title="c1">f</font> is <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> or <font color="Maroon" title="c1">f</font> is <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> or <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E3:7_1_4"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  )
 <b>by </b><i><a class="txt" href="#E1:7_1_4"><i><font color="Green" title="E4">A5</font></i></a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E1:7_1_4"><i><font color="Green" title="E4">A5</font></i></a>, <a class="ref" href="xxreal_3.html#T1" title="XXREAL_3:th.1">XXREAL_3:1</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_5"><b>suppose </b></a><a NAME="E1:7_1_5"/><i><font color="Green" title="E4">A6</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span></span><br/><div class="add"><a NAME="E2:7_1_5"/>
( <font color="Maroon" title="c1">f</font> is <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a> or <font color="Maroon" title="c1">f</font> is <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> or <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E3:7_1_5"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  )
 <b>by </b><i><a class="txt" href="#E1:7_1_5"><i><font color="Green" title="E4">A6</font></i></a>, <a class="ref" href="xxreal_3.html#D2" title="XXREAL_3:def.2">XXREAL_3:def 2</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a>  = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span> <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="xxreal_3.html#K4" title="XXREAL_3:func.4">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E1:7_1_5"><i><font color="Green" title="E4">A6</font></i></a>, <a class="ref" href="xxreal_3.html#T1" title="XXREAL_3:th.1">XXREAL_3:1</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
