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

<b>let </b><font color="Maroon" title="c1">mult1</font>, <font color="Maroon" title="c2">mult2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a>  holds <br/><font color="Maroon" title="c1">mult1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K3" title="CSSPACE:func.3">C_id</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">x</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a>  holds <br/><font color="Maroon" title="c2">mult2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K3" title="CSSPACE:func.3">C_id</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">x</font></span>)</span> ) implies <font color="Maroon" title="c1">mult1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">mult2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:7_1_1"/><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a>  holds <br/><font color="Maroon" title="c1">mult1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K3" title="CSSPACE:func.3">C_id</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">x</font></span>)</span>
 <b>and </b><br/><a NAME="E2:7_1_1"/><i><font color="Green" title="E8">A2</font></i>: 
for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a>  holds <br/><font color="Maroon" title="c2">mult2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="csspace.html#K3" title="CSSPACE:func.3">C_id</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Olive" title="b2">x</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">mult1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">mult2</font></span><br/>

<a NAME="E3:7_1_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a>  holds  <font color="Maroon" title="c1">mult1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">mult2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a>  holds  <font color="Maroon" title="c1">mult1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">z</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">mult2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">z</font>,<font color="Olive" title="b1">x</font></span><br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="csspace.html#K1" title="CSSPACE:func.1">the_set_of_ComplexSequences</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">mult1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">mult2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">x</font></span><br/>



<b>thus </b><font color="Maroon" title="c1">mult1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><a href="csspace.html#K3" title="CSSPACE:func.3">C_id</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="csspace.html#K2" title="CSSPACE:func.2">seq_id</a> <font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="txt" href="csspace.html#E1:7_1_1"><i><font color="Green" title="E7">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">mult2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">z</font>,<font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="txt" href="csspace.html#E2:7_1_1"><i><font color="Green" title="E8">A2</font></i></a></i>
; <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>
<b>hence </b><a NAME="E4:7_1_1"/>
<font color="Maroon" title="c1">mult1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">mult2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
