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<b>consider </b><font color="Maroon" title="c1">A</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>
<a NAME="E2:56_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span> &amp; <font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Olive" title="b5">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_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"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Olive" title="b5">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funcsdom.html#T49" target="_self" title="FUNCSDOM:th.49">Th49</a></i>;<br/>
<a NAME="E3:56_1_1"/><b>then </b>
(  <a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">A</font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp;  <a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">A</font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp;  <a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">A</font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp;  <a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">A</font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp;  <a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">A</font> is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp;  <a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">A</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> &amp;  <a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">A</font> is <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">Algebra-like</a> &amp;  <a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">A</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp;  <a href="funcsdom.html#K13" title="FUNCSDOM:func.13">RAlgebra</a> <font color="Maroon" title="c1">A</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> )
 
<b>by </b><i><a class="ref" href="funcsdom.html#D20" target="_self" title="FUNCSDOM:def.20">Def20</a>, <a class="txt" href="funcsdom.html#E75"><i><font color="Green" title="E32">Lm6</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a>, <a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a>, <a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a>, <a class="ref" href="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a>, <a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:56_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="funcsdom.html#L1" title="FUNCSDOM:struct.1">AlgebraStr</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="funcsdom.html#V1" title="FUNCSDOM:attr.1">strict</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">Algebra-like</a> )
 ; <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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