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

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E1:78"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" 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="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>)</span><br/>

<a NAME="E2:78"/>
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>,<span class="p3">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fcont_1.html#D4" target="_self" title="FCONT_1:def.4">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fcont_1.html#D4" target="_self" title="FCONT_1:def.4">Def3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">b</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<span class="p1">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>;<br/>
<a NAME="E4:78"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="fcont_1.html#T55" target="_self" title="FCONT_1:th.55">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:78"/>
<span class="p1">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K5" title="XCMPLX_0:func.5">"</a></span>)</span>,<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="fcont_1.html#T50" target="_self" title="FCONT_1:th.50">Th27</a>, <a class="ref" href="funct_2.html#T30" title="FUNCT_2:th.30">FUNCT_2:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
