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<b>let </b><font color="Maroon" title="c1">K</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">a1</font>, <font color="Maroon" title="c3">a2</font>, <font color="Maroon" title="c4">a3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a1</font>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font></span>)</span>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font></span>)</span> &amp; the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font></span>)</span>,<font color="Maroon" title="c4">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font></span>)</span>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font></span>)</span> )</span><br/><b>thus </b>the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a1</font>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font></span>)</span> = 
<font color="Maroon" title="c2">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a2</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">a3</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a2</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a3</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></i>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font></span>)</span>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font></span>)</span>

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font></span>)</span>,<font color="Maroon" title="c4">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font></span>)</span>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font></span>)</span></span><br/><b>thus </b>the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c3">a2</font></span>)</span>,<font color="Maroon" title="c4">a3</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">a2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a3</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a3</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">a3</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a></i>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c2">a1</font>,<font color="Maroon" title="c4">a3</font></span>)</span>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a2</font>,<font color="Maroon" title="c4">a3</font></span>)</span>

; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:13"/>
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">K</font> <a href="binop_1.html#R6" title="BINOP_1:pred.6">is_distributive_wrt</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">K</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T23" title="BINOP_1:th.23">BINOP_1:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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