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<span class="kw">let </span><font color="Maroon" title="c1">z</font>, <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">z</font> is  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:52"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
<font color="Maroon" title="c1">z</font> is  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1"/>
( <font color="Maroon" title="c2">x</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="c3">y</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">x</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="c3">y</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="c1">z</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="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">y</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="c1">z</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="c2">x</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="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">z</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="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_1"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
<font color="Maroon" title="c2">x</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1_1_1"/>
( <font color="Maroon" title="c3">y</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="c3">y</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="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A3</font></span>: 
<font color="Maroon" title="c3">y</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font></span><br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> </span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> </span>)</span> = 
<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
<br/>.= 
<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:52_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E1:52_1_1_1_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="52_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_1_1_1_2"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c3">y</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font></span><br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> </span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span>)</span> = 
<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
<br/>.= 
<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:52_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="txt" href="#E1:52_1_1_1_2"><span class="lab"><font color="Green" title="E42">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="52_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:52_1_1_1_3"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<font color="Maroon" title="c3">y</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">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> <span class="kw"> such that </span><br/><a NAME="E3:52_1_1_1_3"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 <span class="kw">and </span><br/><a NAME="E4:52_1_1_1_3"/>
<font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c5">b</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E1:52_1_1_1_3"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E3:52_1_1_1_3"><span class="lab"><font color="Green" title="E43">A6</font></span></a></span>;<br/><a NAME="E6:52_1_1_1_3"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> </span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> = 
<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
<br/>.= 
 <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1_1_1_3"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E6:52_1_1_1_3"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
<br/>.= 
<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1_1_3"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
;<br/><span class="kw">hence </span><a NAME="E8:52_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_1_2"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c3">y</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:52_1_2_1"/>
( <font color="Maroon" title="c2">x</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">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T14" title="XXREAL_0:th.14">XXREAL_0:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:52_1_2_1_1"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<font color="Maroon" title="c2">x</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font></span><br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> </span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> </span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
<br/>.= 
<a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:52_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E1:52_1_2_1_1"><span class="lab"><font color="Green" title="E42">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="52_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:52_1_2_1_2"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
<font color="Maroon" title="c2">x</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font></span><br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
<br/>.= 
<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:52_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E1:52_1_2_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="52_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:52_1_2_1_3"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
<font color="Maroon" title="c2">x</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">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> <span class="kw"> such that </span><br/><a NAME="E3:52_1_2_1_3"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
( <font color="Maroon" title="c1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )
 <span class="kw">and </span><br/><a NAME="E4:52_1_2_1_3"/>
<font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c5">b</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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E1:52_1_2_1_3"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E3:52_1_2_1_3"><span class="lab"><font color="Green" title="E43">A12</font></span></a></span>;<br/><a NAME="E6:52_1_2_1_3"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><a NAME="E7:52_1_2_1_3"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
 <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D3" target="_self" title="XXREAL_3:def.3">Def3</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D3" target="_self" title="XXREAL_3:def.3">Def3</a></span>
<br/>.= 
 <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1_2_1_3"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E6:52_1_2_1_3"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_2_1_3"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E7:52_1_2_1_3"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:52_1_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_3"><span class="kw">suppose </span></a><a NAME="E1:52_1_3"/>
( <font color="Maroon" title="c2">x</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="c3">y</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="c1">z</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c6">c</font> = <font color="Maroon" title="c1">z</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/><span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c5">b</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c4">a</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c5">b</font>

<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>

;<br/><span class="kw">hence </span><a NAME="E4:52_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="52_1_4"><span class="kw">suppose </span></a><a NAME="E1:52_1_4"/><span class="lab"><font color="Green" title="E41">A15</font></span>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">y</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="c1">z</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c1">z</font>, <font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c3">y</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/><a NAME="E3:52_1_4"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
<font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c5">b</font>
 ;<br/><a NAME="E4:52_1_4"/>
( <font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_4"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:52_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:52_1_4"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="ref" href="xxreal_3.html#T13" target="_self" title="XXREAL_3:th.13">Th13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="52_1_5"><span class="kw">suppose </span></a><a NAME="E1:52_1_5"/><span class="lab"><font color="Green" title="E41">A17</font></span>: 
( <font color="Maroon" title="c2">x</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="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">z</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c1">z</font>, <font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c2">x</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/><a NAME="E3:52_1_5"/>
( <font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_5"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> = 
 <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T14" target="_self" title="XXREAL_3:th.14">Th14</a></span>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_5"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="xxreal_3.html#T14" target="_self" title="XXREAL_3:th.14">Th14</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:52_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="52_1_6"><span class="kw">suppose </span></a><a NAME="E1:52_1_6"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  &amp; <font color="Maroon" title="c1">z</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font></span><br/><div class="add"><a NAME="E2:52_1_6"/><span class="kw">then </span>
( <font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a>  &amp; not <font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</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 class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#D2" target="_self" title="XXREAL_3:def.2">Def2</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> = 
 <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T14" target="_self" title="XXREAL_3:th.14">Th14</a></span>
<br/>.= 
<font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_6"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="xxreal_3.html#T14" target="_self" title="XXREAL_3:th.14">Th14</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:52_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z</font> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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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