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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">c1</font>, <font color="Maroon" title="c2">c2</font>, <font color="Maroon" title="c3">c3</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">c2</font>,<font color="Maroon" title="c3">c3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c2">c2</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c3">c3</font></span>)</span> &amp; <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c2">c2</font></span>)</span>,<font color="Maroon" title="c3">c3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c3">c3</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">c2</font>,<font color="Maroon" title="c3">c3</font></span>)</span> )</span><br/><span class="kw">thus </span><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">c2</font>,<font color="Maroon" title="c3">c3</font></span>)</span> = 
<a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">c2</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c3">c3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D3" title="BINOP_2:def.3">BINOP_2:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c1">c1</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c2</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c3">c3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D5" title="BINOP_2:def.5">BINOP_2:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">c1</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c2">c2</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c1</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c3">c3</font></span>)</span>

<br/>.= 
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c1</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c2">c2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">c1</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c3">c3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D3" title="BINOP_2:def.3">BINOP_2:def 3</a></span>
<br/>.= 
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c2">c2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">c1</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c3">c3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D5" title="BINOP_2:def.5">BINOP_2:def 5</a></span>
<br/>.= 
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c2">c2</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c3">c3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D5" title="BINOP_2:def.5">BINOP_2:def 5</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c2">c2</font></span>)</span>,<font color="Maroon" title="c3">c3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c3">c3</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">c2</font>,<font color="Maroon" title="c3">c3</font></span>)</span></span><br/><span class="kw">thus </span><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c2">c2</font></span>)</span>,<font color="Maroon" title="c3">c3</font> = 
<a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c1</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c2">c2</font></span>)</span>,<font color="Maroon" title="c3">c3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D3" title="BINOP_2:def.3">BINOP_2:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">c1</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c2">c2</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c3">c3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D5" title="BINOP_2:def.5">BINOP_2:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">c1</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c3">c3</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">c2</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c3">c3</font></span>)</span>

<br/>.= 
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">c1</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c3">c3</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">c2</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c3">c3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D3" title="BINOP_2:def.3">BINOP_2:def 3</a></span>
<br/>.= 
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c3">c3</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">c2</font> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <font color="Maroon" title="c3">c3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D5" title="BINOP_2:def.5">BINOP_2:def 5</a></span>
<br/>.= 
<a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c1">c1</font>,<font color="Maroon" title="c3">c3</font></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">c2</font>,<font color="Maroon" title="c3">c3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D5" title="BINOP_2:def.5">BINOP_2:def 5</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>
<span class="kw">hence </span><a NAME="E2:77"/>
 <a href="binop_2.html#K29" title="BINOP_2:func.29">multcomplex</a>  <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a>  <a href="binop_2.html#K27" title="BINOP_2:func.27">addcomplex</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T23" title="BINOP_1:th.23">BINOP_1:23</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/>


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