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<b>let </b><font color="Maroon" title="c1">K</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</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">r</font> being  <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b2">M</font>, <font color="Olive" title="b3">N</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b4">m1</font>, <font color="Olive" title="b5">m2</font>, <font color="Olive" title="b6">m</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b2">M</font><br/> for <font color="Olive" title="b7">n1</font>, <font color="Olive" title="b8">n2</font>, <font color="Olive" title="b9">n</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b3">N</font>  st <font color="Olive" title="b2">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Olive" title="b3">N</font> holds <br/>(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b3">N</font> &amp; ( <font color="Olive" title="b4">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">n1</font> &amp; <font color="Olive" title="b5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">n2</font> implies <font color="Olive" title="b4">m1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b8">n2</font> ) &amp; ( <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">n</font> implies <font color="Olive" title="b1">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b9">n</font> ) &amp; ( <font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">n</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b6">m</font> ) &amp; ( <font color="Olive" title="b4">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">n1</font> &amp; <font color="Olive" title="b5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">n2</font> implies <font color="Olive" title="b4">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b8">n2</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b3">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b2">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b3">N</font> &amp; ( <font color="Olive" title="b7">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b8">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">M</font> implies <font color="Olive" title="b7">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b8">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">M</font> ) &amp; ( <font color="Olive" title="b9">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">M</font> implies <font color="Olive" title="b1">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b9">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">M</font> ) &amp; ( <font color="Olive" title="b9">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b9">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">M</font> ) &amp; ( <font color="Olive" title="b7">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b8">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">M</font> implies <font color="Olive" title="b7">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b8">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">M</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">r</font> be   <a href="vectsp_1.html#NM4" title="VECTSP_1:NM.4">Scalar</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"> for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">N</font> being  <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</a> of <font color="Maroon" title="c1">K</font><br/> for <font color="Olive" title="b3">m1</font>, <font color="Olive" title="b4">m2</font>, <font color="Olive" title="b5">m</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b1">M</font><br/> for <font color="Olive" title="b6">n1</font>, <font color="Olive" title="b7">n2</font>, <font color="Olive" title="b8">n</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Olive" title="b2">N</font>  st <font color="Olive" title="b1">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Olive" title="b2">N</font> holds <br/>(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b2">N</font> &amp; ( <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">n1</font> &amp; <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">n2</font> implies <font color="Olive" title="b3">m1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b7">n2</font> ) &amp; ( <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">n</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b8">n</font> ) &amp; ( <font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">n</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b5">m</font> ) &amp; ( <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">n1</font> &amp; <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">n2</font> implies <font color="Olive" title="b3">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b7">n2</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b2">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">N</font> &amp; ( <font color="Olive" title="b6">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b7">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">M</font> implies <font color="Olive" title="b6">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b7">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">M</font> ) &amp; ( <font color="Olive" title="b8">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b8">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">M</font> ) &amp; ( <font color="Olive" title="b8">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b8">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">M</font> ) &amp; ( <font color="Olive" title="b6">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b7">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">M</font> implies <font color="Olive" title="b6">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b7">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">M</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c4">N</font> be   <a href="vectsp_2.html#NM6" title="VECTSP_2:NM.6">LeftMod</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"> for <font color="Olive" title="b1">m1</font>, <font color="Olive" title="b2">m2</font>, <font color="Olive" title="b3">m</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">M</font><br/> for <font color="Olive" title="b4">n1</font>, <font color="Olive" title="b5">n2</font>, <font color="Olive" title="b6">n</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">N</font>  st <font color="Maroon" title="c3">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">N</font> holds <br/>(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">n1</font> &amp; <font color="Olive" title="b2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">n2</font> implies <font color="Olive" title="b1">m1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b5">n2</font> ) &amp; ( <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">n</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b6">n</font> ) &amp; ( <font color="Olive" title="b3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">n</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b3">m</font> ) &amp; ( <font color="Olive" title="b1">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">n1</font> &amp; <font color="Olive" title="b2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">n2</font> implies <font color="Olive" title="b1">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b5">n2</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Olive" title="b4">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b5">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Olive" title="b4">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b5">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Olive" title="b6">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b6">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Olive" title="b6">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b6">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Olive" title="b4">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b5">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Olive" title="b4">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b5">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">m1</font>, <font color="Maroon" title="c6">m2</font>, <font color="Maroon" title="c7">m</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c3">M</font>; <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">n1</font>, <font color="Olive" title="b2">n2</font>, <font color="Olive" title="b3">n</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">N</font>  st <font color="Maroon" title="c3">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">N</font> holds <br/>(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">n2</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b3">n</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">m</font> ) &amp; ( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">n2</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Olive" title="b1">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b2">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Olive" title="b1">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Olive" title="b3">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Olive" title="b3">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Olive" title="b3">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b3">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Olive" title="b1">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b2">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Olive" title="b1">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Olive" title="b2">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) )</span><br/><b>let </b><font color="Maroon" title="c8">n1</font>, <font color="Maroon" title="c9">n2</font>, <font color="Maroon" title="c10">n</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c4">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">N</font> implies (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">m</font> ) &amp; ( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) ) )</span><br/>



















<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c3">M</font> <a href="lmod_6.html#R1" title="LMOD_6:pred.1">c=</a> <font color="Maroon" title="c4">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">m</font> ) &amp; ( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) )</span><br/>

<b>thus </b><a NAME="E2:20"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">m</font> ) &amp; ( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:20"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E2:20_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c11">M'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T19" title="VECTSP_4:th.19">VECTSP_4:19</a></i>;<br/>
<b>hence </b><a NAME="E3:20_1"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font>
 ; <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>thus </b><a NAME="E3:20"/>
( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> ) &amp; ( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">m</font> ) &amp; ( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:20"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">m1'</font> = <font color="Maroon" title="c5">m1</font>, <font color="Maroon" title="c13">m2'</font> = <font color="Maroon" title="c6">m2</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c11">M'</font> ;<br/>
<a NAME="E4:20_2"/>
( <font color="Maroon" title="c12">m1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c13">m2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> )
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:20_2"><i><font color="Green" title="E11">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:20_2"/>
<font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T21" title="VECTSP_4:th.21">VECTSP_4:21</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>thus </b><a NAME="E4:20"/>
( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">m</font> ) &amp; ( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_3"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:20"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">m'</font> = <font color="Maroon" title="c7">m</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c11">M'</font> ;<br/>
<a NAME="E4:20_3"/>
<font color="Maroon" title="c12">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:20_3"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:20_3"/>
<font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</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>thus </b><a NAME="E5:20"/>
( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">m</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_4"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">m</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:20"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">m'</font> = <font color="Maroon" title="c7">m</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c11">M'</font> ;<br/>
<a NAME="E4:20_4"/>
<font color="Maroon" title="c12">m'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:20_4"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:20_4"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T23" title="VECTSP_4:th.23">VECTSP_4:23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:20"/>
( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> implies <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_5"/><i><font color="Green" title="E11">A5</font></i>: 
( <font color="Maroon" title="c5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:20"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">m1'</font> = <font color="Maroon" title="c5">m1</font>, <font color="Maroon" title="c13">m2'</font> = <font color="Maroon" title="c6">m2</font> as   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c11">M'</font> ;<br/>
<a NAME="E4:20_5"/>
( <font color="Maroon" title="c12">m1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> &amp; <font color="Maroon" title="c13">m2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">n2</font> )
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:20_5"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:20_5"/>
<font color="Maroon" title="c5">m1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T24" title="VECTSP_4:th.24">VECTSP_4:24</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>thus </b><a NAME="E7:20"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font> &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_6"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:20"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E2:20_6"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">M'</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T25" title="VECTSP_4:th.25">VECTSP_4:25</a></i>;<br/>
<b>hence </b><a NAME="E3:20_6"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">N</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 ; <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>thus </b><a NAME="E8:20"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_7"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:20"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E2:20_7"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c11">M'</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#T27" title="VECTSP_4:th.27">VECTSP_4:27</a></i>;<br/>
<b>hence </b><a NAME="E3:20_7"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">M</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">N</font>
 ; <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>thus </b><a NAME="E9:20"/>
( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_8"/><i><font color="Green" title="E11">A6</font></i>: 
( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:20"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E3:20_8"/>
( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">M'</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">M'</font> )
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:20_8"><i><font color="Green" title="E11">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:20_8"/>
<font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T28" title="VECTSP_4:th.28">VECTSP_4:28</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>thus </b><a NAME="E10:20"/>
( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) &amp; ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_9"/><i><font color="Green" title="E11">A7</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:20"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E3:20_9"/>
<font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">M'</font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:20_9"><i><font color="Green" title="E11">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:20_9"/>
<font color="Maroon" title="c2">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T29" title="VECTSP_4:th.29">VECTSP_4:29</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>thus </b><a NAME="E11:20"/>
( <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_10"/><i><font color="Green" title="E11">A8</font></i>: 
<font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:20"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E3:20_10"/>
<font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">M'</font>
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:20_10"><i><font color="Green" title="E11">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:20_10"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c10">n</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T30" title="VECTSP_4:th.30">VECTSP_4:30</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>thus </b><a NAME="E12:20"/>
( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> implies <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_11"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_11"/><i><font color="Green" title="E11">A9</font></i>: 
( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">M'</font> = <font color="Maroon" title="c3">M</font> as    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c4">N</font> <b>by </b><i><a class="txt" href="lmod_6.html#E1:20"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="lmod_6.html#D7" target="_self" title="LMOD_6:def.7">Def7</a></i>;<br/>
<a NAME="E3:20_11"/>
( <font color="Maroon" title="c8">n1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">M'</font> &amp; <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">M'</font> )
 
<b>by </b><i><a class="txt" href="lmod_6.html#E1:20_11"><i><font color="Green" title="E11">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:20_11"/>
<font color="Maroon" title="c8">n1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c9">n2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T31" title="VECTSP_4:th.31">VECTSP_4:31</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>


</div>
