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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b3">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  &amp; ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b5">i</font> ) &amp; <font color="Olive" title="b4">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; ( for <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">j</font> &amp; <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b4">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b5">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b4">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b5">i</font> ) holds <br/>( <font color="Olive" title="b3">P</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Olive" title="b4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font><br/> for <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">Q</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b2">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  &amp; ( for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b4">i</font> ) &amp; <font color="Olive" title="b3">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; ( for <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b3">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b4">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b3">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b4">i</font> ) holds <br/>( <font color="Olive" title="b2">P</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Olive" title="b3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  )</span><br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">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">P</font>, <font color="Olive" title="b2">Q</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b1">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b3">i</font> ) &amp; <font color="Olive" title="b2">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b2">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b2">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b3">i</font> ) holds <br/>( <font color="Olive" title="b1">P</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  )</span><br/><b>let </b><font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">Q</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font> ) &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font> ) implies ( <font color="Maroon" title="c4">P</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  ) )</span><br/>







<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E34">A1</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font> ) &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">P</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> &amp; <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  )</span><br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green" title="E35">A40</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:43"/><i><font color="Green" title="E36">A1b</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green" title="E37">B1</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">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="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:43_1"/><i><font color="Green" title="E38">D3</font></i>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_1"><i><font color="Green" title="E37">B1</font></i></a></i>;<br/>
<a NAME="E3:43_1"/><i><font color="Green" title="E39">D4</font></i>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a>, <a class="txt" href="matrix14.html#E2:43_1"><i><font color="Green" title="E38">D3</font></i></a></i>;<br/>
<a NAME="E4:43_1"/><i><font color="Green" title="E40">D5</font></i>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a>, <a class="txt" href="matrix14.html#E3:43_1"><i><font color="Green" title="E39">D4</font></i></a></i>;<br/>
<a NAME="E5:43_1"/><i><font color="Green" title="E41">D6</font></i>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a>, <a class="txt" href="matrix14.html#E4:43_1"><i><font color="Green" title="E40">D5</font></i></a></i>;<br/>
<a NAME="E6:43_1"/><i><font color="Green" title="E42">D8</font></i>: 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#T42" title="VECTSP_1:th.42">VECTSP_1:42</a>, <a class="txt" href="matrix14.html#E5:43_1"><i><font color="Green" title="E41">D6</font></i></a></i>;<br/>
<a NAME="E7:43_1"/>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a>, <a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E6:43_1"><i><font color="Green" title="E42">D8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:43_1"/>
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</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>
<a NAME="E4:43"/><i><font color="Green" title="E37">A2d</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green" title="E38">A2a</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E6:43"/><i><font color="Green" title="E39">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E5:43"><i><font color="Green" title="E38">A2a</font></i></a></i>;<br/>
<a NAME="E7:43"/><i><font color="Green" title="E40">A2b</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<i><font color="Green" title="E41">A20</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<i><font color="Green" title="E42">A21</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<a NAME="E10:43"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> )</span><br/>

<b>assume </b><a NAME="E1:43_4"/><i><font color="Green" title="E43">B1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span><br/>

<a NAME="E2:43_4"/><i><font color="Green" title="E44">B</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:43_4"/><i><font color="Green" title="E45">B2</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="matrix14.html#E1:43_4"><i><font color="Green" title="E43">B1</font></i></a></i>;<br/>
<a NAME="E4:43_4"/><i><font color="Green" title="E46">B3</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="matrix14.html#E4:43"><i><font color="Green" title="E37">A2d</font></i></a>, <a class="txt" href="matrix14.html#E3:43_4"><i><font color="Green" title="E45">B2</font></i></a></i>;<br/>
<a NAME="E5:43_4"/><i><font color="Green" title="E47">A13</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_4"><i><font color="Green" title="E43">B1</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43"><i><font color="Green" title="E40">A2b</font></i></a></i>;<br/>
<a NAME="E6:43_4"/><i><font color="Green" title="E48">A13a</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_4"><i><font color="Green" title="E43">B1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:43_4"/><i><font color="Green" title="E49">B11</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="matrix14.html#E4:43"><i><font color="Green" title="E37">A2d</font></i></a>, <a class="txt" href="matrix14.html#E6:43_4"><i><font color="Green" title="E48">A13a</font></i></a></i>;<br/>
<a NAME="E8:43_4"/><i><font color="Green" title="E50">B92</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E9:43"><i><font color="Green" title="E42">A21</font></i></a>, <a class="txt" href="matrix14.html#E1:43_4"><i><font color="Green" title="E43">B1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<div><i><font color="Green" title="E51">B5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_4_1"/><i><font color="Green" title="E52">C0</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</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="c2">K</font></span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> = 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a>, <a class="txt" href="matrix14.html#E8:43_4"><i><font color="Green" title="E50">B92</font></i></a>, <a class="txt" href="matrix14.html#E5:43_4"><i><font color="Green" title="E47">A13</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E5:43_4"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_4"><i><font color="Green" title="E43">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_4_1"><i><font color="Green" title="E52">C0</font></i></a>, <a class="txt" href="matrix14.html#E6:43_4"><i><font color="Green" title="E48">A13a</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</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><i><font color="Green" title="E52">B6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_4_2"/><i><font color="Green" title="E53">C0</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font></span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> = 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a>, <a class="txt" href="matrix14.html#E8:43_4"><i><font color="Green" title="E50">B92</font></i></a>, <a class="txt" href="matrix14.html#E5:43_4"><i><font color="Green" title="E47">A13</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_4_2"><i><font color="Green" title="E53">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_4"><i><font color="Green" title="E43">B1</font></i></a>, <a class="txt" href="matrix14.html#E5:43_4"><i><font color="Green" title="E47">A13</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_4_3"/><i><font color="Green" title="E53">B20</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</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="c2">K</font></span><br/><a NAME="E2:43_4_3"/><b>then </b><i><font color="Green" title="E54">B19</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_4"><i><font color="Green" title="E43">B1</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:43_4_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">k</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_4"><i><font color="Green" title="E43">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E2:43"><i><font color="Green" title="E35">A40</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">p</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E5:43_4_3"/><i><font color="Green" title="E55">C3</font></i>: 
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,1 = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="matrix14.html#E5:43_4_3"><i><font color="Green" title="E55">C3</font></i></a>, <a class="txt" href="matrix14.html#E2:43_4_3"><i><font color="Green" title="E54">B19</font></i></a>, <a class="txt" href="matrix14.html#E1:43_4"><i><font color="Green" title="E43">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E2:43_4"><i><font color="Green" title="E44">B</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_4_3"><i><font color="Green" title="E53">B20</font></i></a>, <a class="txt" href="matrix14.html#E1:43_4"><i><font color="Green" title="E43">B1</font></i></a>, <a class="txt" href="matrix14.html#E2:43"><i><font color="Green" title="E35">A40</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a></i>
;<br/><b>hence </b><a NAME="E7:43_4_3"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</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="c2">K</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E4:43_4"><i><font color="Green" title="E46">B3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:43_4"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E9:43_4"><i><font color="Green" title="E51">B5</font></i></a>, <a class="txt" href="matrix14.html#E10:43_4"><i><font color="Green" title="E52">B6</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E7:43_4"><i><font color="Green" title="E49">B11</font></i></a>, <a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:43"/><b>then </b><i><font color="Green" title="E43">A3</font></i>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E8:43"><i><font color="Green" title="E41">A20</font></i></a>, <a class="txt" href="matrix14.html#E9:43"><i><font color="Green" title="E42">A21</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E12:43"/><i><font color="Green" title="E44">A4</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a></i>;<br/>
<a NAME="E13:43"/><b>then </b><i><font color="Green" title="E45">A10</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E5:43"><i><font color="Green" title="E38">A2a</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E14:43"/><i><font color="Green" title="E46">A11</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E5:43"><i><font color="Green" title="E38">A2a</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a></i>;<br/>
<a NAME="E15:43"/><i><font color="Green" title="E47">K1</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E14:43"><i><font color="Green" title="E46">A11</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green" title="E48">K3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E5:43"><i><font color="Green" title="E38">A2a</font></i></a></i>
;<br/>
<a NAME="E17:43"/><i><font color="Green" title="E49">K4</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E14:43"><i><font color="Green" title="E46">A11</font></i></a>, <a class="txt" href="matrix14.html#E5:43"><i><font color="Green" title="E38">A2a</font></i></a>, <a class="txt" href="matrix14.html#E6:43"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E18:43"/><i><font color="Green" title="E50">A8d</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E19:43"/><i><font color="Green" title="E51">A8</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E20:43"/><i><font color="Green" title="E52">A12</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></i>;<br/>
<a NAME="E21:43"/><i><font color="Green" title="E53">A18</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<b>then </b><i><font color="Green" title="E54">A13</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="txt" href="matrix14.html#E20:43"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="matrix14.html#E19:43"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="matrix14.html#E4:43"><i><font color="Green" title="E37">A2d</font></i></a>, <a class="txt" href="matrix14.html#E5:43"><i><font color="Green" title="E38">A2a</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="txt" href="matrix14.html#E11:43"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">BB246</a>, <a class="txt" href="matrix14.html#E6:43"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="matrix14.html#E7:43"><i><font color="Green" title="E40">A2b</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E6:43"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a>, <a class="txt" href="matrix14.html#E17:43"><i><font color="Green" title="E49">K4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E16:43"><i><font color="Green" title="E48">K3</font></i></a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>
<b>by </b><i><a class="ref" href="matrix12.html#T3" title="MATRIX12:th.3">MATRIX12:3</a>, <a class="txt" href="matrix14.html#E15:43"><i><font color="Green" title="E47">K1</font></i></a>, <a class="txt" href="matrix14.html#E13:43"><i><font color="Green" title="E45">A10</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a>, <a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a></i>
;<br/>
<a NAME="E23:43"/><i><font color="Green" title="E55">A14</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</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="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</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="c2">K</font> )</span><br/>

<b>assume </b><a NAME="E1:43_7"/><i><font color="Green" title="E56">B1</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</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="c2">K</font></span><br/>

<a NAME="E2:43_7"/><b>then </b><i><font color="Green" title="E57">B20</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:43_7"/><i><font color="Green" title="E58">A22</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c6">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_7"><i><font color="Green" title="E56">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a></i>;<br/>
<i><font color="Green" title="E59">B3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<i><font color="Green" title="E60">B4</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<a NAME="E6:43_7"/><i><font color="Green" title="E61">B5b</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E7:43_7"/><b>then </b><i><font color="Green" title="E62">B5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E5:43_7"><i><font color="Green" title="E60">B4</font></i></a>, <a class="ref" href="matrix14.html#T2" target="_self" title="MATRIX14:th.2">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">j</font>, <font color="Maroon" title="c8">q</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/>
<i><font color="Green" title="E63">A21b</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<a NAME="E10:43_7"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_7_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> )</span><br/>

<b>assume </b><a NAME="E1:43_7_4"/><i><font color="Green" title="E64">C0</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/>

<a NAME="E2:43_7_4"/><i><font color="Green" title="E65">A4</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a></i>;<br/>
<a NAME="E3:43_7_4"/><i><font color="Green" title="E66">B92</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E9:43_7"><i><font color="Green" title="E63">A21b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E4:43_7_4"/><i><font color="Green" title="E67">C12</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="matrix14.html#E4:43"><i><font color="Green" title="E37">A2d</font></i></a></i>;<br/>
<a NAME="E5:43_7_4"/><i><font color="Green" title="E68">A13</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43"><i><font color="Green" title="E40">A2b</font></i></a></i>;<br/>
<i><font color="Green" title="E69">C4</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a>, <a class="txt" href="matrix14.html#E3:43_7_4"><i><font color="Green" title="E66">B92</font></i></a>, <a class="txt" href="matrix14.html#E5:43_7_4"><i><font color="Green" title="E68">A13</font></i></a></i>
;<br/>
<a NAME="E7:43_7_4"/><i><font color="Green" title="E70">A3a</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E8:43_7_4"/><i><font color="Green" title="E71">E1</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43_7_4"><i><font color="Green" title="E70">A3a</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_7_4_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_7_4_2"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> )
 </span><b>by </b><i><a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_7_4_2_1"><b>suppose </b></a><a NAME="E1:43_7_4_2_1"/><i><font color="Green" title="E72">D0</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><i><font color="Green" title="E73">D2</font></i>: <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E4:43_7_4"><i><font color="Green" title="E67">C12</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4_2_1"><i><font color="Green" title="E72">D0</font></i></a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7"><i><font color="Green" title="E56">B1</font></i></a></i>
;<br/><i><font color="Green" title="E74">D1</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_7_4_2_1"><i><font color="Green" title="E72">D0</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="txt" href="matrix14.html#E8:43_7_4"><i><font color="Green" title="E71">E1</font></i></a>, <a class="txt" href="matrix14.html#E6:43_7_4"><i><font color="Green" title="E69">C4</font></i></a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/><i><font color="Green" title="E75">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E6:43_7"><i><font color="Green" title="E61">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_7_4_2_1"><i><font color="Green" title="E72">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7"><i><font color="Green" title="E56">B1</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a></i>
;<br/><a NAME="E5:43_7_4_2_1"/><i><font color="Green" title="E76">K1a</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E5:43_7"><i><font color="Green" title="E60">B4</font></i></a>, <a class="txt" href="matrix14.html#E6:43_7"><i><font color="Green" title="E61">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_7_4_2_1"><i><font color="Green" title="E75">K1</font></i></a></i>;<br/><a NAME="E6:43_7_4_2_1"/>
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a>, <a class="txt" href="matrix14.html#E5:43_7_4_2_1"><i><font color="Green" title="E76">K1a</font></i></a></i>;<br/><b>hence </b><a NAME="E7:43_7_4_2_1"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E2:43_7_4_2_1"><i><font color="Green" title="E73">D2</font></i></a>, <a class="txt" href="matrix14.html#E3:43_7_4_2_1"><i><font color="Green" title="E74">D1</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E5:43_7"><i><font color="Green" title="E60">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_7_4_2_2"><b>suppose </b></a><a NAME="E1:43_7_4_2_2"/><i><font color="Green" title="E72">D0</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><a NAME="E2:43_7_4_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c6">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E1:43_7"><i><font color="Green" title="E56">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E4:43_7_4_2_2"/><i><font color="Green" title="E73">D3</font></i>: 
( <font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/><a NAME="E5:43_7_4_2_2"/><i><font color="Green" title="E74">B</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><i><font color="Green" title="E75">D2</font></i>: <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<font color="Maroon" title="c10">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
<b>by </b><i><a class="txt" href="matrix14.html#E4:43_7_4_2_2"><i><font color="Green" title="E73">D3</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E4:43_7_4"><i><font color="Green" title="E67">C12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4_2_2"><i><font color="Green" title="E72">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_7_4_2_2"><i><font color="Green" title="E73">D3</font></i></a>, <a class="txt" href="matrix14.html#E5:43_7_4_2_2"><i><font color="Green" title="E74">B</font></i></a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_7_4_2_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_7_4_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_7_4_2_2_2_1"/>
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font> or <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_7_4_2_2_2_1_1"><b>suppose </b></a><a NAME="E1:43_7_4_2_2_2_1_1"/><i><font color="Green" title="E76">E0</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E6:43_7_4"><i><font color="Green" title="E69">C4</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E5:43_7"><i><font color="Green" title="E60">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_7_4_2_2"><i><font color="Green" title="E72">D0</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E2:43_7_4"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="txt" href="matrix14.html#E8:43_7_4"><i><font color="Green" title="E71">E1</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><b>then </b><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E5:43_7"><i><font color="Green" title="E60">B4</font></i></a>, <a class="txt" href="matrix14.html#E6:43_7"><i><font color="Green" title="E61">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E6:43_7"><i><font color="Green" title="E61">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_7"><i><font color="Green" title="E56">B1</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4_2_2_2_1_1"><i><font color="Green" title="E76">E0</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:43_7_4_2_2_2_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E6:43_7_4_2_2"><i><font color="Green" title="E75">D2</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_7"><i><font color="Green" title="E56">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4_2_2_2_1_1"><i><font color="Green" title="E76">E0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_7_4_2_2_2_1_2"><b>suppose </b></a><a NAME="E1:43_7_4_2_2_2_1_2"/><i><font color="Green" title="E76">E0</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E6:43_7_4"><i><font color="Green" title="E69">C4</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E5:43_7"><i><font color="Green" title="E60">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_7_4_2_2"><i><font color="Green" title="E72">D0</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E2:43_7_4"><i><font color="Green" title="E65">A4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="txt" href="matrix14.html#E8:43_7_4"><i><font color="Green" title="E71">E1</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><b>then </b><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E5:43_7"><i><font color="Green" title="E60">B4</font></i></a>, <a class="txt" href="matrix14.html#E6:43_7"><i><font color="Green" title="E61">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E6:43_7"><i><font color="Green" title="E61">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4_2_2_2_1_2"><i><font color="Green" title="E76">E0</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:43_7_4_2_2_2_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E6:43_7_4_2_2"><i><font color="Green" title="E75">D2</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_7_4"><i><font color="Green" title="E64">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7_4_2_2_2_1_2"><i><font color="Green" title="E76">E0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:43_7_4_2_2"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</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></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:43_7"/><b>then </b><i><font color="Green" title="E64">B7</font></i>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E4:43_7"><i><font color="Green" title="E59">B3</font></i></a>, <a class="txt" href="matrix14.html#E7:43_7"><i><font color="Green" title="E62">B5</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E12:43_7"/><i><font color="Green" title="E65">A2a</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E13:43_7"/><i><font color="Green" title="E66">K2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E12:43_7"><i><font color="Green" title="E65">A2a</font></i></a></i>;<br/>
<i><font color="Green" title="E67">K3</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E13:43_7"><i><font color="Green" title="E66">K2</font></i></a>, <a class="txt" href="matrix14.html#E1:43_7"><i><font color="Green" title="E56">B1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E2:43_7"><i><font color="Green" title="E57">B20</font></i></a>, <a class="txt" href="matrix14.html#E12:43_7"><i><font color="Green" title="E65">A2a</font></i></a></i>
;<br/>
<i><font color="Green" title="E68">K0</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E13:43_7"><i><font color="Green" title="E66">K2</font></i></a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E13:43"><i><font color="Green" title="E45">A10</font></i></a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="txt" href="matrix14.html#E12:43_7"><i><font color="Green" title="E65">A2a</font></i></a>, <a class="txt" href="matrix14.html#E4:43"><i><font color="Green" title="E37">A2d</font></i></a>, <a class="txt" href="matrix14.html#E3:43_7"><i><font color="Green" title="E58">A22</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T11" target="_self" title="MATRIX14:th.11">BB248</a>, <a class="txt" href="matrix14.html#E5:43_7"><i><font color="Green" title="E60">B4</font></i></a>, <a class="txt" href="matrix14.html#E6:43"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="matrix14.html#E11:43_7"><i><font color="Green" title="E64">B7</font></i></a>, <a class="txt" href="matrix14.html#E6:43_7"><i><font color="Green" title="E61">B5b</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">BB246</a>, <a class="txt" href="matrix14.html#E6:43"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="matrix14.html#E7:43"><i><font color="Green" title="E40">A2b</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="txt" href="matrix14.html#E15:43_7"><i><font color="Green" title="E68">K0</font></i></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E6:43"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a>, <a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E1:43_7"><i><font color="Green" title="E56">B1</font></i></a>, <a class="txt" href="matrix14.html#E6:43"><i><font color="Green" title="E39">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E14:43_7"><i><font color="Green" title="E67">K3</font></i></a>, <a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
;<br/>
<b>hence </b><a NAME="E17:43_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</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="c2">K</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>
<a NAME="E24:43"/><i><font color="Green" title="E56">A15</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</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="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="c2">K</font> )</span><br/>



<b>assume </b><a NAME="E1:43_8"/><i><font color="Green" title="E57">B1</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="c2">K</font></span><br/>

<a NAME="E2:43_8"/><b>then </b><i><font color="Green" title="E58">B20</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:43_8"/><i><font color="Green" title="E59">B2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></i>;<br/>
<a NAME="E4:43_8"/><i><font color="Green" title="E60">A22</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></i>;<br/>
<a NAME="E5:43_8"/><i><font color="Green" title="E61">A23b</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E2:43"><i><font color="Green" title="E35">A40</font></i></a></i>;<br/>
<i><font color="Green" title="E62">B3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<i><font color="Green" title="E63">B4</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<a NAME="E8:43_8"/><i><font color="Green" title="E64">B5b</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E9:43_8"/><b>then </b><i><font color="Green" title="E65">B5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E7:43_8"><i><font color="Green" title="E63">B4</font></i></a>, <a class="ref" href="matrix14.html#T2" target="_self" title="MATRIX14:th.2">Th6</a></i>;<br/>
<i><font color="Green" title="E66">B4b</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_8_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_8_4_1"/>
( <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_1"><b>suppose </b></a><a NAME="E1:43_8_4_1_1"/><i><font color="Green" title="E67">F0</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="c2">K</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c8">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c9">q</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/><a NAME="E3:43_8_4_1_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_8_4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<b>assume </b><a NAME="E1:43_8_4_1_1_1"/><i><font color="Green" title="E68">C0</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<i><font color="Green" title="E69">C10a</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<a NAME="E3:43_8_4_1_1_1"/><i><font color="Green" title="E70">B92b</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="matrix14.html#E2:43_8_4_1_1_1"><i><font color="Green" title="E69">C10a</font></i></a></i>;<br/>
<a NAME="E4:43_8_4_1_1_1"/><i><font color="Green" title="E71">C12</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="matrix14.html#E4:43"><i><font color="Green" title="E37">A2d</font></i></a></i>;<br/>
<a NAME="E5:43_8_4_1_1_1"/><i><font color="Green" title="E72">C2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E4:43_8_4_1_1_1"><i><font color="Green" title="E71">C12</font></i></a></i>;<br/>
<a NAME="E6:43_8_4_1_1_1"/><i><font color="Green" title="E73">A13</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43"><i><font color="Green" title="E40">A2b</font></i></a></i>;<br/>
<i><font color="Green" title="E74">C4</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a>, <a class="txt" href="matrix14.html#E3:43_8_4_1_1_1"><i><font color="Green" title="E70">B92b</font></i></a>, <a class="txt" href="matrix14.html#E6:43_8_4_1_1_1"><i><font color="Green" title="E73">A13</font></i></a></i>
;<br/>
<a NAME="E8:43_8_4_1_1_1"/><i><font color="Green" title="E75">A3a</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E9:43_8_4_1_1_1"/><i><font color="Green" title="E76">E1</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E8:43_8_4_1_1_1"><i><font color="Green" title="E75">A3a</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_1_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_8_4_1_1_1_3"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> )
 </span><b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_1_1_3_1"><b>suppose </b></a><a NAME="E1:43_8_4_1_1_1_3_1"/><i><font color="Green" title="E77">D0</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_8_4_1_1_1_3_1"/><i><font color="Green" title="E78">K3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="txt" href="matrix14.html#E2:43_8_4_1_1_1"><i><font color="Green" title="E69">C10a</font></i></a>, <a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>;<br/><i><font color="Green" title="E79">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E8:43_8"><i><font color="Green" title="E64">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_1_1_3_1"><i><font color="Green" title="E77">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1"><i><font color="Green" title="E67">F0</font></i></a></i>
;<br/><i><font color="Green" title="E80">K2</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E7:43_8_4_1_1_1"><i><font color="Green" title="E74">C4</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E2:43_8_4_1_1_1_3_1"><i><font color="Green" title="E78">K3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_1_1_3_1"><i><font color="Green" title="E77">D0</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="matrix14.html#E9:43_8_4_1_1_1"><i><font color="Green" title="E76">E1</font></i></a></i>
;<br/><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E7:43_8"><i><font color="Green" title="E63">B4</font></i></a>, <a class="txt" href="matrix14.html#E8:43_8"><i><font color="Green" title="E64">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_8_4_1_1_1_3_1"><i><font color="Green" title="E80">K2</font></i></a>, <a class="txt" href="matrix14.html#E3:43_8_4_1_1_1_3_1"><i><font color="Green" title="E79">K1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/><b>hence </b><a NAME="E6:43_8_4_1_1_1_3_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1"><i><font color="Green" title="E67">F0</font></i></a>, <a class="txt" href="matrix14.html#E5:43_8_4_1_1_1"><i><font color="Green" title="E72">C2</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1_3_1"><i><font color="Green" title="E77">D0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_1_1_3_2"><b>suppose </b></a><a NAME="E1:43_8_4_1_1_1_3_2"/><i><font color="Green" title="E77">D0</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_8_4_1_1_1_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E4:43_8_4_1_1_1_3_2"/><i><font color="Green" title="E78">D3</font></i>: 
( <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/><a NAME="E5:43_8_4_1_1_1_3_2"/><i><font color="Green" title="E79">K3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="txt" href="matrix14.html#E2:43_8_4_1_1_1"><i><font color="Green" title="E69">C10a</font></i></a>, <a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>;<br/><a NAME="E6:43_8_4_1_1_1_3_2"/><i><font color="Green" title="E80">B</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E7:43_8_4_1_1_1_3_2"/><i><font color="Green" title="E81">D2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1_3_2"><i><font color="Green" title="E77">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_8_4_1_1_1_3_2"><i><font color="Green" title="E78">D3</font></i></a>, <a class="txt" href="matrix14.html#E6:43_8_4_1_1_1_3_2"><i><font color="Green" title="E80">B</font></i></a>, <a class="txt" href="matrix14.html#E5:43_8_4_1_1_1"><i><font color="Green" title="E72">C2</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_8_4_1_1_1_3_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_1_1_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_8_4_1_1_1_3_2_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_1_1_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_8_4_1_1_1_3_2_1_1_1"/><i><font color="Green" title="E82">E0</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><i><font color="Green" title="E83">E4</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_1_1_3_2"><i><font color="Green" title="E77">D0</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="txt" href="matrix14.html#E9:43_8_4_1_1_1"><i><font color="Green" title="E76">E1</font></i></a>, <a class="txt" href="matrix14.html#E7:43_8_4_1_1_1"><i><font color="Green" title="E74">C4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><i><font color="Green" title="E84">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E8:43_8"><i><font color="Green" title="E64">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1_3_2_1_1_1"><i><font color="Green" title="E82">E0</font></i></a></i>
;<br/><a NAME="E4:43_8_4_1_1_1_3_2_1_1_1"/><i><font color="Green" title="E85">K2</font></i>: 
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E7:43_8"><i><font color="Green" title="E63">B4</font></i></a>, <a class="txt" href="matrix14.html#E8:43_8"><i><font color="Green" title="E64">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="txt" href="matrix14.html#E3:43_8_4_1_1_1_3_2_1_1_1"><i><font color="Green" title="E84">K1</font></i></a></i>;<br/><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a>, <a class="txt" href="matrix14.html#E4:43_8_4_1_1_1_3_2_1_1_1"><i><font color="Green" title="E85">K2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E5:43_8_4_1_1_1_3_2"><i><font color="Green" title="E79">K3</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:43_8_4_1_1_1_3_2_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E7:43_8_4_1_1_1_3_2"><i><font color="Green" title="E81">D2</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1_3_2_1_1_1"><i><font color="Green" title="E82">E0</font></i></a>, <a class="txt" href="matrix14.html#E2:43_8_4_1_1_1_3_2_1_1_1"><i><font color="Green" title="E83">E4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_1_1_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_8_4_1_1_1_3_2_1_1_2"/><i><font color="Green" title="E82">E0</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_8_4_1_1_1_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E83">E2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E7:43_8_4_1_1_1_3_2"><i><font color="Green" title="E81">D2</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a></i>;<br/><i><font color="Green" title="E84">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E8:43_8"><i><font color="Green" title="E64">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1_3_2_1_1_2"><i><font color="Green" title="E82">E0</font></i></a></i>
;<br/><a NAME="E4:43_8_4_1_1_1_3_2_1_1_2"/><i><font color="Green" title="E85">K2</font></i>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43_8"><i><font color="Green" title="E63">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a></i>;<br/><i><font color="Green" title="E86">E3</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_1_1_3_2"><i><font color="Green" title="E77">D0</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="txt" href="matrix14.html#E6:43_8_4_1_1_1"><i><font color="Green" title="E73">A13</font></i></a>, <a class="txt" href="matrix14.html#E7:43_8_4_1_1_1"><i><font color="Green" title="E74">C4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><a NAME="E6:43_8_4_1_1_1_3_2_1_1_2"/>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E5:43_8_4_1_1_1_3_2_1_1_2"><i><font color="Green" title="E86">E3</font></i></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E7:43_8"><i><font color="Green" title="E63">B4</font></i></a>, <a class="txt" href="matrix14.html#E8:43_8"><i><font color="Green" title="E64">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_1_1"><i><font color="Green" title="E68">C0</font></i></a>, <a class="txt" href="matrix14.html#E3:43_8_4_1_1_1_3_2_1_1_2"><i><font color="Green" title="E84">K1</font></i></a>, <a class="txt" href="matrix14.html#E4:43_8_4_1_1_1_3_2_1_1_2"><i><font color="Green" title="E85">K2</font></i></a></i>;<br/><b>hence </b><a NAME="E7:43_8_4_1_1_1_3_2_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E2:43_8_4_1_1_1_3_2_1_1_2"><i><font color="Green" title="E83">E2</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:43_8_4_1_1_1_3_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</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></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:43_8_4_1_1"/><b>then </b><i><font color="Green" title="E68">B7</font></i>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E6:43_8"><i><font color="Green" title="E62">B3</font></i></a>, <a class="txt" href="matrix14.html#E9:43_8"><i><font color="Green" title="E65">B5</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/><a NAME="E5:43_8_4_1_1"/><i><font color="Green" title="E69">A2a</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><a NAME="E6:43_8_4_1_1"/><i><font color="Green" title="E70">K2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E5:43_8_4_1_1"><i><font color="Green" title="E69">A2a</font></i></a></i>;<br/><i><font color="Green" title="E71">K3</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E6:43_8_4_1_1"><i><font color="Green" title="E70">K2</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E2:43_8"><i><font color="Green" title="E58">B20</font></i></a>, <a class="txt" href="matrix14.html#E5:43_8_4_1_1"><i><font color="Green" title="E69">A2a</font></i></a></i>
;<br/><i><font color="Green" title="E72">K0</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E6:43_8_4_1_1"><i><font color="Green" title="E70">K2</font></i></a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E13:43"><i><font color="Green" title="E45">A10</font></i></a></i>
;<br/><a NAME="E9:43_8_4_1_1"/><i><font color="Green" title="E73">B18</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a></i>;<br/><i><font color="Green" title="E74">B28</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="txt" href="matrix14.html#E5:43_8_4_1_1"><i><font color="Green" title="E69">A2a</font></i></a>, <a class="txt" href="matrix14.html#E4:43"><i><font color="Green" title="E37">A2d</font></i></a>, <a class="txt" href="matrix14.html#E4:43_8"><i><font color="Green" title="E60">A22</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T11" target="_self" title="MATRIX14:th.11">BB248</a>, <a class="txt" href="matrix14.html#E7:43_8"><i><font color="Green" title="E63">B4</font></i></a>, <a class="txt" href="matrix14.html#E6:43_8_4_1_1"><i><font color="Green" title="E70">K2</font></i></a>, <a class="txt" href="matrix14.html#E4:43_8_4_1_1"><i><font color="Green" title="E68">B7</font></i></a>, <a class="txt" href="matrix14.html#E8:43_8"><i><font color="Green" title="E64">B5b</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">BB246</a>, <a class="txt" href="matrix14.html#E6:43_8_4_1_1"><i><font color="Green" title="E70">K2</font></i></a>, <a class="txt" href="matrix14.html#E7:43"><i><font color="Green" title="E40">A2b</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="txt" href="matrix14.html#E8:43_8_4_1_1"><i><font color="Green" title="E72">K0</font></i></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E6:43_8_4_1_1"><i><font color="Green" title="E70">K2</font></i></a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E7:43_8_4_1_1"><i><font color="Green" title="E71">K3</font></i></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a>, <a class="txt" href="matrix14.html#E6:43_8_4_1_1"><i><font color="Green" title="E70">K2</font></i></a></i>
;<br/><b>consider </b><font color="Maroon" title="c10">p1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E12:43_8_4_1_1"/><i><font color="Green" title="E75">A97</font></i>: 
( <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a>, <a class="txt" href="matrix14.html#E3:43_8"><i><font color="Green" title="E59">B2</font></i></a></i>;<br/><a NAME="E13:43_8_4_1_1"/>
<font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E12:43_8_4_1_1"><i><font color="Green" title="E75">A97</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a></i>;<br/><a NAME="E14:43_8_4_1_1"/><b>then </b><i><font color="Green" title="E76">A98</font></i>: 
<font color="Maroon" title="c10">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</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="c2">K</font>
 <b>by </b><i><a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E16:43_8_4_1_1"/><i><font color="Green" title="E77">A99</font></i>: 
( <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a>, <a class="txt" href="matrix14.html#E5:43_8"><i><font color="Green" title="E61">A23b</font></i></a></i>;<br/><a NAME="E17:43_8_4_1_1"/><i><font color="Green" title="E78">K2</font></i>: 
<font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E16:43_8_4_1_1"><i><font color="Green" title="E77">A99</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E10:43_8_4_1_1"><i><font color="Green" title="E74">B28</font></i></a>, <a class="txt" href="matrix14.html#E16:43_8_4_1_1"><i><font color="Green" title="E77">A99</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E9:43_8_4_1_1"><i><font color="Green" title="E73">B18</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a>, <a class="txt" href="matrix14.html#E17:43_8_4_1_1"><i><font color="Green" title="E78">K2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E12:43_8_4_1_1"><i><font color="Green" title="E75">A97</font></i></a>, <a class="txt" href="matrix14.html#E14:43_8_4_1_1"><i><font color="Green" title="E76">A98</font></i></a>, <a class="ref" href="rlvect_1.html#T25" title="RLVECT_1:th.25">RLVECT_1:25</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/><b>hence </b><a NAME="E19:43_8_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="c2">K</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_2"><b>suppose </b></a><a NAME="E1:43_8_4_1_2"/><i><font color="Green" title="E67">F1</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="c2">K</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c8">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/><a NAME="E3:43_8_4_1_2"/><i><font color="Green" title="E68">F2</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_8_4_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<b>assume </b><a NAME="E1:43_8_4_1_2_1"/><i><font color="Green" title="E69">C0</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<a NAME="E2:43_8_4_1_2_1"/><i><font color="Green" title="E70">B92</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E9:43"><i><font color="Green" title="E42">A21</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_2_1"><i><font color="Green" title="E69">C0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:43_8_4_1_2_1"/><i><font color="Green" title="E71">C12</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_2_1"><i><font color="Green" title="E69">C0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="matrix14.html#E4:43"><i><font color="Green" title="E37">A2d</font></i></a></i>;<br/>
<a NAME="E4:43_8_4_1_2_1"/><i><font color="Green" title="E72">A2b</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E5:43_8_4_1_2_1"/><i><font color="Green" title="E73">A13</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_2_1"><i><font color="Green" title="E69">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E4:43_8_4_1_2_1"><i><font color="Green" title="E72">A2b</font></i></a></i>;<br/>
<a NAME="E6:43_8_4_1_2_1"/><i><font color="Green" title="E74">C4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a>, <a class="txt" href="matrix14.html#E2:43_8_4_1_2_1"><i><font color="Green" title="E70">B92</font></i></a>, <a class="txt" href="matrix14.html#E5:43_8_4_1_2_1"><i><font color="Green" title="E73">A13</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_8_4_1_2_1_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> )
 </span><b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_2_1"><i><font color="Green" title="E69">C0</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_8_4_1_2_1_1_1"/><i><font color="Green" title="E75">D0</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><i><font color="Green" title="E76">D2</font></i>: <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E3:43_8_4_1_2_1"><i><font color="Green" title="E71">C12</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_2_1_1_1"><i><font color="Green" title="E75">D0</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_2"><i><font color="Green" title="E67">F1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>
;<br/><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E6:43_8_4_1_2_1"><i><font color="Green" title="E74">C4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_2_1_1_1"><i><font color="Green" title="E75">D0</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_2_1"><i><font color="Green" title="E69">C0</font></i></a>, <a class="txt" href="matrix14.html#E5:43_8_4_1_2_1"><i><font color="Green" title="E73">A13</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
;<br/><b>hence </b><a NAME="E4:43_8_4_1_2_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E2:43_8_4_1_2_1_1_1"><i><font color="Green" title="E76">D2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_8_4_1_2_1_1_2"/><i><font color="Green" title="E75">D0</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_8_4_1_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_2_1"><i><font color="Green" title="E69">C0</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E4:43_8_4_1_2_1_1_2"/><i><font color="Green" title="E76">D3</font></i>: 
( <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/><a NAME="E5:43_8_4_1_2_1_1_2"/><i><font color="Green" title="E77">B</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><i><font color="Green" title="E78">D2</font></i>: <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E3:43_8_4_1_2_1"><i><font color="Green" title="E71">C12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_2_1_1_2"><i><font color="Green" title="E75">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_2_1"><i><font color="Green" title="E69">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_8_4_1_2_1_1_2"><i><font color="Green" title="E76">D3</font></i></a>, <a class="txt" href="matrix14.html#E5:43_8_4_1_2_1_1_2"><i><font color="Green" title="E77">B</font></i></a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_8_4_1_2_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_2_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_8_4_1_2_1_1_2_2_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_2_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:43_8_4_1_2_1_1_2_2_1_1"/><i><font color="Green" title="E79">E0</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_8_4_1_2_1_1_2_2_1_1"/><i><font color="Green" title="E80">A13</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_2_1"><i><font color="Green" title="E69">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E4:43_8_4_1_2_1"><i><font color="Green" title="E72">A2b</font></i></a></i>;<br/><a NAME="E3:43_8_4_1_2_1_1_2_2_1_1"/><i><font color="Green" title="E81">A13a</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_2_1"><i><font color="Green" title="E69">C0</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span> = 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_8_4_1_2_1_1_2_2_1_1"><i><font color="Green" title="E80">A13</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_2_1_1_2"><i><font color="Green" title="E75">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_2_1"><i><font color="Green" title="E69">C0</font></i></a>, <a class="txt" href="matrix14.html#E3:43_8_4_1_2_1_1_2_2_1_1"><i><font color="Green" title="E81">A13a</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><b>hence </b><a NAME="E5:43_8_4_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_2_1_1_2_2_1_1"><i><font color="Green" title="E79">E0</font></i></a>, <a class="txt" href="matrix14.html#E6:43_8_4_1_2_1_1_2"><i><font color="Green" title="E78">D2</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_2_1"><i><font color="Green" title="E69">C0</font></i></a>, <a class="txt" href="matrix14.html#E6:43_8_4_1_2_1"><i><font color="Green" title="E74">C4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_8_4_1_2_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:43_8_4_1_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><b>hence </b><a NAME="E2:43_8_4_1_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_8_4_1_2"><i><font color="Green" title="E67">F1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8_4_1_2_1_1_2"><i><font color="Green" title="E75">D0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:43_8_4_1_2_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</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></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:43_8_4_1_2"/><i><font color="Green" title="E69">A2x</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E5:43"><i><font color="Green" title="E38">A2a</font></i></a></i>;<br/><i><font color="Green" title="E70">K0</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E4:43_8_4_1_2"><i><font color="Green" title="E69">A2x</font></i></a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E13:43"><i><font color="Green" title="E45">A10</font></i></a></i>
;<br/><a NAME="E6:43_8_4_1_2"/><i><font color="Green" title="E71">B18</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a></i>;<br/><i><font color="Green" title="E72">B28</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="txt" href="matrix14.html#E5:43"><i><font color="Green" title="E38">A2a</font></i></a>, <a class="txt" href="matrix14.html#E4:43"><i><font color="Green" title="E37">A2d</font></i></a>, <a class="txt" href="matrix14.html#E4:43_8"><i><font color="Green" title="E60">A22</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="txt" href="matrix14.html#E3:43_8_4_1_2"><i><font color="Green" title="E68">F2</font></i></a>, <a class="txt" href="matrix14.html#E6:43_8"><i><font color="Green" title="E62">B3</font></i></a>, <a class="txt" href="matrix14.html#E10:43_8"><i><font color="Green" title="E66">B4b</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">BB246</a>, <a class="txt" href="matrix14.html#E4:43_8_4_1_2"><i><font color="Green" title="E69">A2x</font></i></a>, <a class="txt" href="matrix14.html#E7:43"><i><font color="Green" title="E40">A2b</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E5:43_8_4_1_2"><i><font color="Green" title="E70">K0</font></i></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E4:43_8_4_1_2"><i><font color="Green" title="E69">A2x</font></i></a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a></i>
;<br/><b>consider </b><font color="Maroon" title="c10">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E9:43_8_4_1_2"/><i><font color="Green" title="E73">A99</font></i>: 
( <font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a>, <a class="txt" href="matrix14.html#E5:43_8"><i><font color="Green" title="E61">A23b</font></i></a></i>;<br/><a NAME="E10:43_8_4_1_2"/>
<font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E9:43_8_4_1_2"><i><font color="Green" title="E73">A99</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E7:43_8_4_1_2"><i><font color="Green" title="E72">B28</font></i></a>, <a class="txt" href="matrix14.html#E9:43_8_4_1_2"><i><font color="Green" title="E73">A99</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E6:43_8_4_1_2"><i><font color="Green" title="E71">B18</font></i></a>, <a class="txt" href="matrix14.html#E1:43_8"><i><font color="Green" title="E57">B1</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/> <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><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:43_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="c2">K</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>
<a NAME="E25:43"/><i><font color="Green" title="E57">A85</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font> )</span><br/>



<b>assume </b><a NAME="E1:43_9"/><i><font color="Green" title="E58">B1</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span><br/>

<a NAME="E2:43_9"/><b>then </b><i><font color="Green" title="E59">B20</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:43_9"/><i><font color="Green" title="E60">B2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></i>;<br/>
<a NAME="E4:43_9"/><i><font color="Green" title="E61">A22</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></i>;<br/>
<a NAME="E5:43_9"/><i><font color="Green" title="E62">A23b</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E2:43"><i><font color="Green" title="E35">A40</font></i></a></i>;<br/>
<i><font color="Green" title="E63">B3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<i><font color="Green" title="E64">B4</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<a NAME="E8:43_9"/><i><font color="Green" title="E65">B5b</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E9:43_9"/><b>then </b><i><font color="Green" title="E66">B5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E7:43_9"><i><font color="Green" title="E64">B4</font></i></a>, <a class="ref" href="matrix14.html#T2" target="_self" title="MATRIX14:th.2">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c9">q</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/>
<a NAME="E11:43_9"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_9_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<b>assume </b><a NAME="E1:43_9_3"/><i><font color="Green" title="E67">C0</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<a NAME="E2:43_9_3"/><b>then </b><i><font color="Green" title="E68">C10</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<i><font color="Green" title="E69">C10a</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<i><font color="Green" title="E70">A21b</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<a NAME="E5:43_9_3"/><i><font color="Green" title="E71">B92b</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E4:43_9_3"><i><font color="Green" title="E70">A21b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:43_9_3"/><i><font color="Green" title="E72">C12</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="matrix14.html#E4:43"><i><font color="Green" title="E37">A2d</font></i></a></i>;<br/>
<a NAME="E7:43_9_3"/><i><font color="Green" title="E73">C2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E6:43_9_3"><i><font color="Green" title="E72">C12</font></i></a></i>;<br/>
<a NAME="E8:43_9_3"/><i><font color="Green" title="E74">A13</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43"><i><font color="Green" title="E40">A2b</font></i></a></i>;<br/>
<i><font color="Green" title="E75">C4</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a>, <a class="txt" href="matrix14.html#E5:43_9_3"><i><font color="Green" title="E71">B92b</font></i></a>, <a class="txt" href="matrix14.html#E8:43_9_3"><i><font color="Green" title="E74">A13</font></i></a></i>
;<br/>
<a NAME="E10:43_9_3"/><i><font color="Green" title="E76">A3a</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E11:43_9_3"/><i><font color="Green" title="E77">E1</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E10:43_9_3"><i><font color="Green" title="E76">A3a</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_9_3_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_9_3_4"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> )
 </span><b>by </b><i><a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_9_3_4_1"><b>suppose </b></a><a NAME="E1:43_9_3_4_1"/><i><font color="Green" title="E78">D0</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_9_3_4_1"/><i><font color="Green" title="E79">K3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a>, <a class="txt" href="matrix14.html#E3:43_9_3"><i><font color="Green" title="E69">C10a</font></i></a>, <a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>;<br/><i><font color="Green" title="E80">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E8:43_9"><i><font color="Green" title="E65">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_9_3_4_1"><i><font color="Green" title="E78">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a></i>
;<br/><i><font color="Green" title="E81">K2</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E9:43_9_3"><i><font color="Green" title="E75">C4</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E2:43_9_3_4_1"><i><font color="Green" title="E79">K3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_9_3_4_1"><i><font color="Green" title="E78">D0</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="matrix14.html#E11:43_9_3"><i><font color="Green" title="E77">E1</font></i></a></i>
;<br/><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E7:43_9"><i><font color="Green" title="E64">B4</font></i></a>, <a class="txt" href="matrix14.html#E8:43_9"><i><font color="Green" title="E65">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_9_3_4_1"><i><font color="Green" title="E81">K2</font></i></a>, <a class="txt" href="matrix14.html#E3:43_9_3_4_1"><i><font color="Green" title="E80">K1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
;<br/><b>hence </b><a NAME="E6:43_9_3_4_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a>, <a class="txt" href="matrix14.html#E7:43_9_3"><i><font color="Green" title="E73">C2</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3_4_1"><i><font color="Green" title="E78">D0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_9_3_4_2"><b>suppose </b></a><a NAME="E1:43_9_3_4_2"/><i><font color="Green" title="E78">D0</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_9_3_4_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E18:43"><i><font color="Green" title="E50">A8d</font></i></a>, <a class="txt" href="matrix14.html#E2:43_9"><i><font color="Green" title="E59">B20</font></i></a>, <a class="txt" href="matrix14.html#E2:43_9_3"><i><font color="Green" title="E68">C10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E4:43_9_3_4_2"/><i><font color="Green" title="E79">D3</font></i>: 
( <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/><a NAME="E5:43_9_3_4_2"/><i><font color="Green" title="E80">B</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:43_9_3_4_2"/><i><font color="Green" title="E81">D2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3_4_2"><i><font color="Green" title="E78">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_9_3_4_2"><i><font color="Green" title="E79">D3</font></i></a>, <a class="txt" href="matrix14.html#E5:43_9_3_4_2"><i><font color="Green" title="E80">B</font></i></a>, <a class="txt" href="matrix14.html#E7:43_9_3"><i><font color="Green" title="E73">C2</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_9_3_4_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_9_3_4_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_9_3_4_2_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_9_3_4_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_9_3_4_2_1_1_1"/><i><font color="Green" title="E82">E0</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_9_3_4_2_1_1_1"/><b>then </b><i><font color="Green" title="E83">E2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</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="c2">K</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E6:43_9_3_4_2"><i><font color="Green" title="E81">D2</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a></i>;<br/><i><font color="Green" title="E84">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E8:43_9"><i><font color="Green" title="E65">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3_4_2_1_1_1"><i><font color="Green" title="E82">E0</font></i></a></i>
;<br/><a NAME="E4:43_9_3_4_2_1_1_1"/><i><font color="Green" title="E85">K2</font></i>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43_9"><i><font color="Green" title="E64">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a></i>;<br/><i><font color="Green" title="E86">E3</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_9_3_4_2"><i><font color="Green" title="E78">D0</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a>, <a class="txt" href="matrix14.html#E8:43_9_3"><i><font color="Green" title="E74">A13</font></i></a>, <a class="txt" href="matrix14.html#E9:43_9_3"><i><font color="Green" title="E75">C4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><a NAME="E6:43_9_3_4_2_1_1_1"/>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E5:43_9_3_4_2_1_1_1"><i><font color="Green" title="E86">E3</font></i></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E7:43_9"><i><font color="Green" title="E64">B4</font></i></a>, <a class="txt" href="matrix14.html#E8:43_9"><i><font color="Green" title="E65">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a>, <a class="txt" href="matrix14.html#E3:43_9_3_4_2_1_1_1"><i><font color="Green" title="E84">K1</font></i></a>, <a class="txt" href="matrix14.html#E4:43_9_3_4_2_1_1_1"><i><font color="Green" title="E85">K2</font></i></a></i>;<br/><b>hence </b><a NAME="E7:43_9_3_4_2_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E2:43_9_3_4_2_1_1_1"><i><font color="Green" title="E83">E2</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_9_3_4_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_9_3_4_2_1_1_2"/><i><font color="Green" title="E82">E0</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_9_3_4_2_1_1_2"/><b>then </b><i><font color="Green" title="E83">E2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E6:43_9_3_4_2"><i><font color="Green" title="E81">D2</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a></i>;<br/><i><font color="Green" title="E84">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E8:43_9"><i><font color="Green" title="E65">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_9_3_4_2_1_1_2"><i><font color="Green" title="E82">E0</font></i></a></i>
;<br/><a NAME="E4:43_9_3_4_2_1_1_2"/><i><font color="Green" title="E85">K2</font></i>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43_9"><i><font color="Green" title="E64">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a></i>;<br/><i><font color="Green" title="E86">E3</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_9_3_4_2"><i><font color="Green" title="E78">D0</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a>, <a class="txt" href="matrix14.html#E8:43_9_3"><i><font color="Green" title="E74">A13</font></i></a>, <a class="txt" href="matrix14.html#E9:43_9_3"><i><font color="Green" title="E75">C4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><a NAME="E6:43_9_3_4_2_1_1_2"/>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E5:43_9_3_4_2_1_1_2"><i><font color="Green" title="E86">E3</font></i></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E7:43_9"><i><font color="Green" title="E64">B4</font></i></a>, <a class="txt" href="matrix14.html#E8:43_9"><i><font color="Green" title="E65">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9_3"><i><font color="Green" title="E67">C0</font></i></a>, <a class="txt" href="matrix14.html#E3:43_9_3_4_2_1_1_2"><i><font color="Green" title="E84">K1</font></i></a>, <a class="txt" href="matrix14.html#E4:43_9_3_4_2_1_1_2"><i><font color="Green" title="E85">K2</font></i></a></i>;<br/><b>hence </b><a NAME="E7:43_9_3_4_2_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E2:43_9_3_4_2_1_1_2"><i><font color="Green" title="E83">E2</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:43_9_3_4_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</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></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E12:43_9"/><b>then </b><i><font color="Green" title="E67">B7</font></i>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E6:43_9"><i><font color="Green" title="E63">B3</font></i></a>, <a class="txt" href="matrix14.html#E9:43_9"><i><font color="Green" title="E66">B5</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E13:43_9"/><i><font color="Green" title="E68">A2a</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E14:43_9"/><i><font color="Green" title="E69">K2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E13:43_9"><i><font color="Green" title="E68">A2a</font></i></a></i>;<br/>
<i><font color="Green" title="E70">K0</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E14:43_9"><i><font color="Green" title="E69">K2</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E2:43_9"><i><font color="Green" title="E59">B20</font></i></a>, <a class="txt" href="matrix14.html#E13:43_9"><i><font color="Green" title="E68">A2a</font></i></a></i>
;<br/>
<i><font color="Green" title="E71">K3</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E14:43_9"><i><font color="Green" title="E69">K2</font></i></a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E13:43"><i><font color="Green" title="E45">A10</font></i></a></i>
;<br/>
<a NAME="E17:43_9"/><i><font color="Green" title="E72">B18</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a></i>;<br/>
<i><font color="Green" title="E73">B28</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="txt" href="matrix14.html#E13:43_9"><i><font color="Green" title="E68">A2a</font></i></a>, <a class="txt" href="matrix14.html#E4:43"><i><font color="Green" title="E37">A2d</font></i></a>, <a class="txt" href="matrix14.html#E4:43_9"><i><font color="Green" title="E61">A22</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T11" target="_self" title="MATRIX14:th.11">BB248</a>, <a class="txt" href="matrix14.html#E7:43_9"><i><font color="Green" title="E64">B4</font></i></a>, <a class="txt" href="matrix14.html#E14:43_9"><i><font color="Green" title="E69">K2</font></i></a>, <a class="txt" href="matrix14.html#E12:43_9"><i><font color="Green" title="E67">B7</font></i></a>, <a class="txt" href="matrix14.html#E8:43_9"><i><font color="Green" title="E65">B5b</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">BB246</a>, <a class="txt" href="matrix14.html#E14:43_9"><i><font color="Green" title="E69">K2</font></i></a>, <a class="txt" href="matrix14.html#E7:43"><i><font color="Green" title="E40">A2b</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="txt" href="matrix14.html#E16:43_9"><i><font color="Green" title="E71">K3</font></i></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E14:43_9"><i><font color="Green" title="E69">K2</font></i></a>, <a class="txt" href="matrix14.html#E12:43"><i><font color="Green" title="E44">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E15:43_9"><i><font color="Green" title="E70">K0</font></i></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a>, <a class="txt" href="matrix14.html#E14:43_9"><i><font color="Green" title="E69">K2</font></i></a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c10">p1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E20:43_9"/><i><font color="Green" title="E74">A97</font></i>: 
( <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a>, <a class="txt" href="matrix14.html#E3:43_9"><i><font color="Green" title="E60">B2</font></i></a></i>;<br/>
<a NAME="E21:43_9"/>
<font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E20:43_9"><i><font color="Green" title="E74">A97</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a></i>;<br/>
<a NAME="E22:43_9"/><b>then </b><i><font color="Green" title="E75">A98</font></i>: 
<font color="Maroon" title="c10">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E24:43_9"/><i><font color="Green" title="E76">A99</font></i>: 
( <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a>, <a class="txt" href="matrix14.html#E5:43_9"><i><font color="Green" title="E62">A23b</font></i></a></i>;<br/>
<a NAME="E25:43_9"/>
<font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E24:43_9"><i><font color="Green" title="E76">A99</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E24:43_9"><i><font color="Green" title="E76">A99</font></i></a>, <a class="txt" href="matrix14.html#E18:43_9"><i><font color="Green" title="E73">B28</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E17:43_9"><i><font color="Green" title="E72">B18</font></i></a>, <a class="txt" href="matrix14.html#E1:43_9"><i><font color="Green" title="E58">B1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E20:43_9"><i><font color="Green" title="E74">A97</font></i></a>, <a class="txt" href="matrix14.html#E22:43_9"><i><font color="Green" title="E75">A98</font></i></a>, <a class="ref" href="rlvect_1.html#T25" title="RLVECT_1:th.25">RLVECT_1:25</a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/> <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>
<a NAME="E26:43"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> )</span><br/>

<b>assume </b><a NAME="E1:43_10"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/>

<a NAME="E2:43_10"/><b>then </b><i><font color="Green" title="E58">B2</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="matrix14.html#E21:43"><i><font color="Green" title="E53">A18</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:43_10"/><b>then </b><i><font color="Green" title="E59">B3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:43_10"/><i><font color="Green" title="E60">B4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_10"><i><font color="Green" title="E58">B2</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">i0</font> = <font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c9">j0</font> = <font color="Maroon" title="c7">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_10_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> or 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> )
 </span><b>by </b><i><a class="txt" href="matrix14.html#E3:43_10"><i><font color="Green" title="E59">B3</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_1"><b>suppose </b></a><a NAME="E1:43_10_1_1"/><i><font color="Green" title="E61">C0</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_10_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_10_1_1_1_1"/>
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_10_1_1_1_1_1"/><i><font color="Green" title="E62">D0</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><i><font color="Green" title="E63">D1</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<b>by </b><i><a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">BB300</a>, <a class="txt" href="matrix14.html#E3:43_10"><i><font color="Green" title="E59">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_10"><i><font color="Green" title="E60">B4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_10_1_1_1_1_1"><i><font color="Green" title="E62">D0</font></i></a>, <a class="txt" href="matrix14.html#E3:43_10"><i><font color="Green" title="E59">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_10"><i><font color="Green" title="E60">B4</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i0</font>,<font color="Maroon" title="c9">j0</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_10_1_1_1_1_1"><i><font color="Green" title="E63">D1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_10_1_1"><i><font color="Green" title="E61">C0</font></i></a>, <a class="txt" href="matrix14.html#E3:43_10"><i><font color="Green" title="E59">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_10"><i><font color="Green" title="E60">B4</font></i></a>, <a class="txt" href="matrix14.html#E24:43"><i><font color="Green" title="E56">A15</font></i></a>, <a class="txt" href="matrix14.html#E1:43_10_1_1_1_1_1"><i><font color="Green" title="E62">D0</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_10_1_1_1_1_2"/><i><font color="Green" title="E62">D0</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><i><font color="Green" title="E63">D1</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<b>by </b><i><a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">BB300</a>, <a class="txt" href="matrix14.html#E3:43_10"><i><font color="Green" title="E59">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_10"><i><font color="Green" title="E60">B4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_10_1_1_1_1_2"><i><font color="Green" title="E62">D0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_10"><i><font color="Green" title="E60">B4</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i0</font>,<font color="Maroon" title="c9">j0</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_10_1_1_1_1_2"><i><font color="Green" title="E63">D1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_10_1_1"><i><font color="Green" title="E61">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_10"><i><font color="Green" title="E60">B4</font></i></a>, <a class="txt" href="matrix14.html#E25:43"><i><font color="Green" title="E57">A85</font></i></a>, <a class="txt" href="matrix14.html#E1:43_10_1_1_1_1_2"><i><font color="Green" title="E62">D0</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:43_10_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_2"><b>suppose </b></a><a NAME="E1:43_10_1_2"/><i><font color="Green" title="E61">C0</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_10_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_10_1_2_1_1"/>
( <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_10_1_2_1_1_1"/><i><font color="Green" title="E62">D0</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><i><font color="Green" title="E63">D1</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<b>by </b><i><a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">BB300</a>, <a class="txt" href="matrix14.html#E3:43_10"><i><font color="Green" title="E59">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_10"><i><font color="Green" title="E60">B4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E3:43_10"><i><font color="Green" title="E59">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_10"><i><font color="Green" title="E60">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_10_1_2_1_1_1"><i><font color="Green" title="E62">D0</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i0</font>,<font color="Maroon" title="c9">j0</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_10_1_2_1_1_1"><i><font color="Green" title="E63">D1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_10_1_2"><i><font color="Green" title="E61">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_10"><i><font color="Green" title="E60">B4</font></i></a>, <a class="txt" href="matrix14.html#E23:43"><i><font color="Green" title="E55">A14</font></i></a>, <a class="txt" href="matrix14.html#E1:43_10_1_2_1_1_1"><i><font color="Green" title="E62">D0</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_10_1_2_1_1_2"/><i><font color="Green" title="E62">D9</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><a NAME="E2:43_10_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E63">D0</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E4:43_10"><i><font color="Green" title="E60">B4</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="txt" href="matrix14.html#E1:43_10_1_2"><i><font color="Green" title="E61">C0</font></i></a></i>;<br/><i><font color="Green" title="E64">D1</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<b>by </b><i><a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">BB300</a>, <a class="txt" href="matrix14.html#E3:43_10"><i><font color="Green" title="E59">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_10"><i><font color="Green" title="E60">B4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_10_1_2_1_1_2"><i><font color="Green" title="E63">D0</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E4:43_10"><i><font color="Green" title="E60">B4</font></i></a></i>
;<br/><b>thus </b><a NAME="E4:43_10_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E3:43_10_1_2_1_1_2"><i><font color="Green" title="E64">D1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_10_1_2"><i><font color="Green" title="E61">C0</font></i></a>, <a class="txt" href="matrix14.html#E22:43"><i><font color="Green" title="E54">A13</font></i></a>, <a class="txt" href="matrix14.html#E1:43_10_1_2_1_1_2"><i><font color="Green" title="E62">D9</font></i></a>, <a class="txt" href="matrix14.html#E4:43_10"><i><font color="Green" title="E60">B4</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:43_10_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E27:43"/><b>then </b><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a></i>;<br/>
<a NAME="E28:43"/><i><font color="Green" title="E59">A2d</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E29:43"/><i><font color="Green" title="E60">A2a</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E30:43"/><i><font color="Green" title="E61">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E29:43"><i><font color="Green" title="E60">A2a</font></i></a></i>;<br/>
<a NAME="E31:43"/><i><font color="Green" title="E62">A2b</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<i><font color="Green" title="E63">A20</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<i><font color="Green" title="E64">A21</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<a NAME="E34:43"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> )</span><br/>

<a NAME="E1:43_13"/><i><font color="Green" title="E65">B</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:43_13"/><i><font color="Green" title="E66">B1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span><br/>

<a NAME="E3:43_13"/><b>then </b><i><font color="Green" title="E67">B32</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:43_13"/><b>then </b><i><font color="Green" title="E68">B3</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="matrix14.html#E28:43"><i><font color="Green" title="E59">A2d</font></i></a></i>;<br/>
<a NAME="E5:43_13"/><i><font color="Green" title="E69">B92</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E33:43"><i><font color="Green" title="E64">A21</font></i></a>, <a class="txt" href="matrix14.html#E2:43_13"><i><font color="Green" title="E66">B1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:43_13"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_13"><i><font color="Green" title="E66">B1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:43_13"/><b>then </b><i><font color="Green" title="E70">B11</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="matrix14.html#E28:43"><i><font color="Green" title="E59">A2d</font></i></a></i>;<br/>
<a NAME="E8:43_13"/><i><font color="Green" title="E71">A13</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_13"><i><font color="Green" title="E66">B1</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E31:43"><i><font color="Green" title="E62">A2b</font></i></a></i>;<br/>
<a NAME="E9:43_13"/><i><font color="Green" title="E72">A13a</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_13"><i><font color="Green" title="E66">B1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<div><i><font color="Green" title="E73">B5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_13_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_13_1"/><i><font color="Green" title="E74">C0</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</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="c2">K</font></span><br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a>, <a class="txt" href="matrix14.html#E5:43_13"><i><font color="Green" title="E69">B92</font></i></a>, <a class="txt" href="matrix14.html#E8:43_13"><i><font color="Green" title="E71">A13</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E8:43_13"><i><font color="Green" title="E71">A13</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E2:43_13"><i><font color="Green" title="E66">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_13_1"><i><font color="Green" title="E74">C0</font></i></a>, <a class="txt" href="matrix14.html#E9:43_13"><i><font color="Green" title="E72">A13a</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</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><i><font color="Green" title="E74">B6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_13_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_13_2"/><i><font color="Green" title="E75">C0</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span><br/><a NAME="E2:43_13_2"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E3:43_13"><i><font color="Green" title="E67">B32</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="txt" href="matrix14.html#E33:43"><i><font color="Green" title="E64">A21</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a>, <a class="txt" href="matrix14.html#E8:43_13"><i><font color="Green" title="E71">A13</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_13_2"><i><font color="Green" title="E75">C0</font></i></a>, <a class="txt" href="matrix14.html#E2:43_13"><i><font color="Green" title="E66">B1</font></i></a>, <a class="txt" href="matrix14.html#E8:43_13"><i><font color="Green" title="E71">A13</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<b>by </b><i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
;<br/> <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><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_13_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_13_3"/><i><font color="Green" title="E75">B20</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</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="c2">K</font></span><br/><a NAME="E2:43_13_3"/><b>then </b><i><font color="Green" title="E76">B19</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E2:43_13"><i><font color="Green" title="E66">B1</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:43_13_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">k</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E2:43_13"><i><font color="Green" title="E66">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E2:43"><i><font color="Green" title="E35">A40</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">p</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E5:43_13_3"/><i><font color="Green" title="E77">C3</font></i>: 
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,1 = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="matrix14.html#E5:43_13_3"><i><font color="Green" title="E77">C3</font></i></a>, <a class="txt" href="matrix14.html#E2:43_13_3"><i><font color="Green" title="E76">B19</font></i></a>, <a class="txt" href="matrix14.html#E2:43_13"><i><font color="Green" title="E66">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_13"><i><font color="Green" title="E65">B</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_13_3"><i><font color="Green" title="E75">B20</font></i></a>, <a class="txt" href="matrix14.html#E2:43_13"><i><font color="Green" title="E66">B1</font></i></a>, <a class="txt" href="matrix14.html#E2:43"><i><font color="Green" title="E35">A40</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a></i>
;<br/><b>hence </b><a NAME="E7:43_13_3"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</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="c2">K</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E4:43_13"><i><font color="Green" title="E68">B3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:43_13"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E10:43_13"><i><font color="Green" title="E73">B5</font></i></a>, <a class="txt" href="matrix14.html#E11:43_13"><i><font color="Green" title="E74">B6</font></i></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E7:43_13"><i><font color="Green" title="E70">B11</font></i></a>, <a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E35:43"/><b>then </b><i><font color="Green" title="E65">A3</font></i>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E32:43"><i><font color="Green" title="E63">A20</font></i></a>, <a class="txt" href="matrix14.html#E33:43"><i><font color="Green" title="E64">A21</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E36:43"/><i><font color="Green" title="E66">A4</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a></i>;<br/>
<a NAME="E37:43"/><b>then </b><i><font color="Green" title="E67">A10</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">Q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E29:43"><i><font color="Green" title="E60">A2a</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E38:43"/><i><font color="Green" title="E68">A11</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E29:43"><i><font color="Green" title="E60">A2a</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a></i>;<br/>
<a NAME="E39:43"/><i><font color="Green" title="E69">K1</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E38:43"><i><font color="Green" title="E68">A11</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green" title="E70">K3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E29:43"><i><font color="Green" title="E60">A2a</font></i></a></i>
;<br/>
<a NAME="E41:43"/><i><font color="Green" title="E71">K4</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E38:43"><i><font color="Green" title="E68">A11</font></i></a>, <a class="txt" href="matrix14.html#E29:43"><i><font color="Green" title="E60">A2a</font></i></a>, <a class="txt" href="matrix14.html#E30:43"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E42:43"/><i><font color="Green" title="E72">A8d</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E43:43"/><i><font color="Green" title="E73">A8</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<a NAME="E44:43"/><i><font color="Green" title="E74">A12</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></i>;<br/>
<a NAME="E45:43"/><i><font color="Green" title="E75">A18</font></i>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<b>then </b><i><font color="Green" title="E76">A13</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="txt" href="matrix14.html#E44:43"><i><font color="Green" title="E74">A12</font></i></a>, <a class="txt" href="matrix14.html#E43:43"><i><font color="Green" title="E73">A8</font></i></a>, <a class="txt" href="matrix14.html#E28:43"><i><font color="Green" title="E59">A2d</font></i></a>, <a class="txt" href="matrix14.html#E29:43"><i><font color="Green" title="E60">A2a</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="txt" href="matrix14.html#E35:43"><i><font color="Green" title="E65">A3</font></i></a>, <a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">BB246</a>, <a class="txt" href="matrix14.html#E30:43"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="matrix14.html#E31:43"><i><font color="Green" title="E62">A2b</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E30:43"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<b>by </b><i><a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a>, <a class="txt" href="matrix14.html#E41:43"><i><font color="Green" title="E71">K4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E40:43"><i><font color="Green" title="E70">K3</font></i></a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>
<b>by </b><i><a class="ref" href="matrix12.html#T3" title="MATRIX12:th.3">MATRIX12:3</a>, <a class="txt" href="matrix14.html#E39:43"><i><font color="Green" title="E69">K1</font></i></a>, <a class="txt" href="matrix14.html#E37:43"><i><font color="Green" title="E67">A10</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a>, <a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a></i>
;<br/>
<a NAME="E47:43"/><i><font color="Green" title="E77">A14</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</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="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_16"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</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="c2">K</font> )</span><br/>

<b>assume </b><a NAME="E1:43_16"/><i><font color="Green" title="E78">B1</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</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="c2">K</font></span><br/>

<a NAME="E2:43_16"/><b>then </b><i><font color="Green" title="E79">B20</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:43_16"/><i><font color="Green" title="E80">A22</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c6">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_16"><i><font color="Green" title="E78">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a></i>;<br/>
<i><font color="Green" title="E81">B3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<i><font color="Green" title="E82">B4</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<i><font color="Green" title="E83">B4d</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<a NAME="E7:43_16"/><i><font color="Green" title="E84">B5b</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E8:43_16"/><b>then </b><i><font color="Green" title="E85">B5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E5:43_16"><i><font color="Green" title="E82">B4</font></i></a>, <a class="ref" href="matrix14.html#T2" target="_self" title="MATRIX14:th.2">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">j</font>, <font color="Maroon" title="c8">q</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/>
<a NAME="E10:43_16"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_16_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> )</span><br/>

<b>assume </b><a NAME="E1:43_16_4"/><i><font color="Green" title="E86">C0</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/>

<a NAME="E2:43_16_4"/><b>then </b><i><font color="Green" title="E87">C10b</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E6:43_16"><i><font color="Green" title="E83">B4d</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<i><font color="Green" title="E88">C10a</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<a NAME="E4:43_16_4"/><i><font color="Green" title="E89">C12</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="matrix14.html#E28:43"><i><font color="Green" title="E59">A2d</font></i></a></i>;<br/>
<a NAME="E5:43_16_4"/><i><font color="Green" title="E90">C2</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c6">j</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E4:43_16_4"><i><font color="Green" title="E89">C12</font></i></a></i>;<br/>
<a NAME="E6:43_16_4"/><i><font color="Green" title="E91">A2b</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E7:43_16_4"/><i><font color="Green" title="E92">A13</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E6:43_16_4"><i><font color="Green" title="E91">A2b</font></i></a></i>;<br/>
<i><font color="Green" title="E93">C4</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a>, <a class="txt" href="matrix14.html#E2:43_16_4"><i><font color="Green" title="E87">C10b</font></i></a>, <a class="txt" href="matrix14.html#E7:43_16_4"><i><font color="Green" title="E92">A13</font></i></a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_16_4_3"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> )
 </span><b>by </b><i><a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_3_1"><b>suppose </b></a><a NAME="E1:43_16_4_3_1"/><i><font color="Green" title="E94">D0</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><a NAME="E2:43_16_4_3_1"/><b>then </b><i><font color="Green" title="E95">D2</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E3:43"><i><font color="Green" title="E36">A1b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16"><i><font color="Green" title="E78">B1</font></i></a>, <a class="txt" href="matrix14.html#E5:43_16_4"><i><font color="Green" title="E90">C2</font></i></a></i>;<br/><a NAME="E3:43_16_4_3_1"/><i><font color="Green" title="E96">K3</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="txt" href="matrix14.html#E3:43_16_4"><i><font color="Green" title="E88">C10a</font></i></a>, <a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a>, <a class="txt" href="matrix14.html#E1:43_16_4_3_1"><i><font color="Green" title="E94">D0</font></i></a></i>;<br/><i><font color="Green" title="E97">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43_16"><i><font color="Green" title="E84">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4_3_1"><i><font color="Green" title="E94">D0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_16_4_3_1"><i><font color="Green" title="E94">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16"><i><font color="Green" title="E78">B1</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a></i>
;<br/><i><font color="Green" title="E98">D1</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E8:43_16_4"><i><font color="Green" title="E93">C4</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E3:43_16_4_3_1"><i><font color="Green" title="E96">K3</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4_3_1"><i><font color="Green" title="E94">D0</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_16_4_3_1"><i><font color="Green" title="E94">D0</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="matrix14.html#E7:43_16_4"><i><font color="Green" title="E92">A13</font></i></a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
;<br/><a NAME="E6:43_16_4_3_1"/>
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E5:43_16_4_3_1"><i><font color="Green" title="E98">D1</font></i></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E5:43_16"><i><font color="Green" title="E82">B4</font></i></a>, <a class="txt" href="matrix14.html#E7:43_16"><i><font color="Green" title="E84">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_16_4_3_1"><i><font color="Green" title="E97">K1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4_3_1"><i><font color="Green" title="E94">D0</font></i></a></i>;<br/><b>hence </b><a NAME="E7:43_16_4_3_1"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E2:43_16_4_3_1"><i><font color="Green" title="E95">D2</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a>, <a class="txt" href="matrix14.html#E1:43_16_4_3_1"><i><font color="Green" title="E94">D0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_3_2"><b>suppose </b></a><a NAME="E1:43_16_4_3_2"/><i><font color="Green" title="E94">D0</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><a NAME="E2:43_16_4_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c6">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16"><i><font color="Green" title="E78">B1</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E4:43_16_4_3_2"/><i><font color="Green" title="E95">D3</font></i>: 
( <font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/><a NAME="E5:43_16_4_3_2"/><i><font color="Green" title="E96">B</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:43_16_4_3_2"/><i><font color="Green" title="E97">D2</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4_3_2"><i><font color="Green" title="E94">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_16_4_3_2"><i><font color="Green" title="E95">D3</font></i></a>, <a class="txt" href="matrix14.html#E5:43_16_4_3_2"><i><font color="Green" title="E96">B</font></i></a>, <a class="txt" href="matrix14.html#E5:43_16_4"><i><font color="Green" title="E90">C2</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_16_4_3_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_16_4_3_2_1_1"/>
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font> or <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_16_4_3_2_1_1_1"/><i><font color="Green" title="E98">E0</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><a NAME="E2:43_16_4_3_2_1_1_1"/><b>then </b><i><font color="Green" title="E99">E2</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</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="c2">K</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E6:43_16_4_3_2"><i><font color="Green" title="E97">D2</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_16"><i><font color="Green" title="E78">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a></i>;<br/><a NAME="E3:43_16_4_3_2_1_1_1"/><i><font color="Green" title="E100">A3a</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/><a NAME="E4:43_16_4_3_2_1_1_1"/><i><font color="Green" title="E101">E1</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E3:43_16_4_3_2_1_1_1"><i><font color="Green" title="E100">A3a</font></i></a></i>;<br/><i><font color="Green" title="E102">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43_16"><i><font color="Green" title="E84">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_16"><i><font color="Green" title="E78">B1</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4_3_2_1_1_1"><i><font color="Green" title="E98">E0</font></i></a></i>
;<br/><a NAME="E6:43_16_4_3_2_1_1_1"/><i><font color="Green" title="E103">K2</font></i>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E5:43_16"><i><font color="Green" title="E82">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a></i>;<br/><i><font color="Green" title="E104">E3</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_16_4_3_2"><i><font color="Green" title="E94">D0</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_16_4_3_2_1_1_1"><i><font color="Green" title="E101">E1</font></i></a>, <a class="txt" href="matrix14.html#E8:43_16_4"><i><font color="Green" title="E93">C4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><a NAME="E8:43_16_4_3_2_1_1_1"/>
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E7:43_16_4_3_2_1_1_1"><i><font color="Green" title="E104">E3</font></i></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E5:43_16"><i><font color="Green" title="E82">B4</font></i></a>, <a class="txt" href="matrix14.html#E7:43_16"><i><font color="Green" title="E84">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="txt" href="matrix14.html#E5:43_16_4_3_2_1_1_1"><i><font color="Green" title="E102">K1</font></i></a>, <a class="txt" href="matrix14.html#E6:43_16_4_3_2_1_1_1"><i><font color="Green" title="E103">K2</font></i></a></i>;<br/><b>hence </b><a NAME="E9:43_16_4_3_2_1_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E2:43_16_4_3_2_1_1_1"><i><font color="Green" title="E99">E2</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_16_4_3_2_1_1_2"/><i><font color="Green" title="E98">E0</font></i>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><a NAME="E2:43_16_4_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E99">E2</font></i>: 
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E6:43_16_4_3_2"><i><font color="Green" title="E97">D2</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a></i>;<br/><a NAME="E3:43_16_4_3_2_1_1_2"/><i><font color="Green" title="E100">A3a</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/><a NAME="E4:43_16_4_3_2_1_1_2"/><i><font color="Green" title="E101">E1</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E3:43_16_4_3_2_1_1_2"><i><font color="Green" title="E100">A3a</font></i></a></i>;<br/><i><font color="Green" title="E102">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43_16"><i><font color="Green" title="E84">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4_3_2_1_1_2"><i><font color="Green" title="E98">E0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_16"><i><font color="Green" title="E78">B1</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_16_4_3_2_1_1_2"><i><font color="Green" title="E98">E0</font></i></a></i>
;<br/><a NAME="E6:43_16_4_3_2_1_1_2"/><i><font color="Green" title="E103">K2</font></i>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E5:43_16"><i><font color="Green" title="E82">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4_3_2_1_1_2"><i><font color="Green" title="E98">E0</font></i></a></i>;<br/><i><font color="Green" title="E104">E3</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_16_4_3_2"><i><font color="Green" title="E94">D0</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_16_4_3_2_1_1_2"><i><font color="Green" title="E101">E1</font></i></a>, <a class="txt" href="matrix14.html#E8:43_16_4"><i><font color="Green" title="E93">C4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4_3_2_1_1_2"><i><font color="Green" title="E98">E0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><a NAME="E8:43_16_4_3_2_1_1_2"/>
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E5:43_16"><i><font color="Green" title="E82">B4</font></i></a>, <a class="txt" href="matrix14.html#E7:43_16"><i><font color="Green" title="E84">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4"><i><font color="Green" title="E86">C0</font></i></a>, <a class="txt" href="matrix14.html#E7:43_16_4_3_2_1_1_2"><i><font color="Green" title="E104">E3</font></i></a>, <a class="txt" href="matrix14.html#E5:43_16_4_3_2_1_1_2"><i><font color="Green" title="E102">K1</font></i></a>, <a class="txt" href="matrix14.html#E6:43_16_4_3_2_1_1_2"><i><font color="Green" title="E103">K2</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16_4_3_2_1_1_2"><i><font color="Green" title="E98">E0</font></i></a></i>;<br/><b>hence </b><a NAME="E9:43_16_4_3_2_1_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E2:43_16_4_3_2_1_1_2"><i><font color="Green" title="E99">E2</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:43_16_4_3_2"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</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></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:43_16"/><b>then </b><i><font color="Green" title="E86">B7</font></i>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E4:43_16"><i><font color="Green" title="E81">B3</font></i></a>, <a class="txt" href="matrix14.html#E8:43_16"><i><font color="Green" title="E85">B5</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E12:43_16"/><i><font color="Green" title="E87">K1</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_16"><i><font color="Green" title="E78">B1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:43_16"/><i><font color="Green" title="E88">B8</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">Q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="matrix14.html#E29:43"><i><font color="Green" title="E60">A2a</font></i></a>, <a class="txt" href="matrix14.html#E12:43_16"><i><font color="Green" title="E87">K1</font></i></a></i>;<br/>
<i><font color="Green" title="E89">K2</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E12:43_16"><i><font color="Green" title="E87">K1</font></i></a>, <a class="txt" href="matrix14.html#E30:43"><i><font color="Green" title="E61">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E13:43_16"><i><font color="Green" title="E88">B8</font></i></a></i>
;<br/>
<a NAME="E15:43_16"/><i><font color="Green" title="E90">K2a</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E29:43"><i><font color="Green" title="E60">A2a</font></i></a></i>;<br/>
<a NAME="E16:43_16"/><i><font color="Green" title="E91">K3</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E15:43_16"><i><font color="Green" title="E90">K2a</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16"><i><font color="Green" title="E78">B1</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="txt" href="matrix14.html#E29:43"><i><font color="Green" title="E60">A2a</font></i></a>, <a class="txt" href="matrix14.html#E28:43"><i><font color="Green" title="E59">A2d</font></i></a>, <a class="txt" href="matrix14.html#E3:43_16"><i><font color="Green" title="E80">A22</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T11" target="_self" title="MATRIX14:th.11">BB248</a>, <a class="txt" href="matrix14.html#E5:43_16"><i><font color="Green" title="E82">B4</font></i></a>, <a class="txt" href="matrix14.html#E30:43"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="matrix14.html#E11:43_16"><i><font color="Green" title="E86">B7</font></i></a>, <a class="txt" href="matrix14.html#E7:43_16"><i><font color="Green" title="E84">B5b</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">BB246</a>, <a class="txt" href="matrix14.html#E30:43"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="matrix14.html#E31:43"><i><font color="Green" title="E62">A2b</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="txt" href="matrix14.html#E14:43_16"><i><font color="Green" title="E89">K2</font></i></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E30:43"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a>, <a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E30:43"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="matrix14.html#E1:43_16"><i><font color="Green" title="E78">B1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E29:43"><i><font color="Green" title="E60">A2a</font></i></a>, <a class="txt" href="matrix14.html#E2:43_16"><i><font color="Green" title="E79">B20</font></i></a>, <a class="txt" href="matrix14.html#E16:43_16"><i><font color="Green" title="E91">K3</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
;<br/>
<b>hence </b><a NAME="E18:43_16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</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="c2">K</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>
<a NAME="E48:43"/><i><font color="Green" title="E78">A15</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</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="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="c2">K</font> )</span><br/>



<b>assume </b><a NAME="E1:43_17"/><i><font color="Green" title="E79">B1</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="c2">K</font></span><br/>

<a NAME="E2:43_17"/><b>then </b><i><font color="Green" title="E80">B20</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:43_17"/><i><font color="Green" title="E81">B2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></i>;<br/>
<a NAME="E4:43_17"/><i><font color="Green" title="E82">A22</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></i>;<br/>
<a NAME="E5:43_17"/><i><font color="Green" title="E83">A23b</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a></i>;<br/>
<i><font color="Green" title="E84">B3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<i><font color="Green" title="E85">B4</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<i><font color="Green" title="E86">B4d</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<a NAME="E9:43_17"/><i><font color="Green" title="E87">B5b</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E10:43_17"/><b>then </b><i><font color="Green" title="E88">B5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E7:43_17"><i><font color="Green" title="E85">B4</font></i></a>, <a class="ref" href="matrix14.html#T2" target="_self" title="MATRIX14:th.2">Th6</a></i>;<br/>
<i><font color="Green" title="E89">B4b</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_17_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_17_5_1"/>
( <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_1"><b>suppose </b></a><a NAME="E1:43_17_5_1_1"/><i><font color="Green" title="E90">F0</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="c2">K</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c8">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c9">q</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/><a NAME="E3:43_17_5_1_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_17_5_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<b>assume </b><a NAME="E1:43_17_5_1_1_1"/><i><font color="Green" title="E91">C0</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<a NAME="E2:43_17_5_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:43_17_5_1_1_1"/><b>then </b><i><font color="Green" title="E92">C10b</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E8:43_17"><i><font color="Green" title="E86">B4d</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:43_17_5_1_1_1"/><i><font color="Green" title="E93">C12</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="matrix14.html#E28:43"><i><font color="Green" title="E59">A2d</font></i></a></i>;<br/>
<a NAME="E5:43_17_5_1_1_1"/><i><font color="Green" title="E94">C2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E4:43_17_5_1_1_1"><i><font color="Green" title="E93">C12</font></i></a></i>;<br/>
<a NAME="E6:43_17_5_1_1_1"/><i><font color="Green" title="E95">A2b</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E7:43_17_5_1_1_1"/><i><font color="Green" title="E96">A13</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E6:43_17_5_1_1_1"><i><font color="Green" title="E95">A2b</font></i></a></i>;<br/>
<i><font color="Green" title="E97">C4</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a>, <a class="txt" href="matrix14.html#E3:43_17_5_1_1_1"><i><font color="Green" title="E92">C10b</font></i></a>, <a class="txt" href="matrix14.html#E7:43_17_5_1_1_1"><i><font color="Green" title="E96">A13</font></i></a></i>
;<br/>
<a NAME="E9:43_17_5_1_1_1"/><i><font color="Green" title="E98">A3a</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E10:43_17_5_1_1_1"/><i><font color="Green" title="E99">E1</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E9:43_17_5_1_1_1"><i><font color="Green" title="E98">A3a</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_17_5_1_1_1_2"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> )
 </span><b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_1_1_2_1"><b>suppose </b></a><a NAME="E1:43_17_5_1_1_1_2_1"/><i><font color="Green" title="E100">D0</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><i><font color="Green" title="E101">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E9:43_17"><i><font color="Green" title="E87">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_1_1_2_1"><i><font color="Green" title="E100">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1"><i><font color="Green" title="E90">F0</font></i></a></i>
;<br/><a NAME="E3:43_17_5_1_1_1_2_1"/><i><font color="Green" title="E102">K2</font></i>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43_17"><i><font color="Green" title="E85">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a></i>;<br/><a NAME="E4:43_17_5_1_1_1_2_1"/><i><font color="Green" title="E103">D1</font></i>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_1_1_2_1"><i><font color="Green" title="E100">D0</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="matrix14.html#E7:43_17_5_1_1_1"><i><font color="Green" title="E96">A13</font></i></a>, <a class="txt" href="matrix14.html#E8:43_17_5_1_1_1"><i><font color="Green" title="E97">C4</font></i></a></i>;<br/><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E7:43_17"><i><font color="Green" title="E85">B4</font></i></a>, <a class="txt" href="matrix14.html#E9:43_17"><i><font color="Green" title="E87">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_17_5_1_1_1_2_1"><i><font color="Green" title="E103">D1</font></i></a>, <a class="txt" href="matrix14.html#E2:43_17_5_1_1_1_2_1"><i><font color="Green" title="E101">K1</font></i></a>, <a class="txt" href="matrix14.html#E3:43_17_5_1_1_1_2_1"><i><font color="Green" title="E102">K2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
;<br/><b>hence </b><a NAME="E6:43_17_5_1_1_1_2_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E3:43"><i><font color="Green" title="E36">A1b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1"><i><font color="Green" title="E90">F0</font></i></a>, <a class="txt" href="matrix14.html#E5:43_17_5_1_1_1"><i><font color="Green" title="E94">C2</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1_2_1"><i><font color="Green" title="E100">D0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_1_1_2_2"><b>suppose </b></a><a NAME="E1:43_17_5_1_1_1_2_2"/><i><font color="Green" title="E100">D0</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_17_5_1_1_1_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E4:43_17_5_1_1_1_2_2"/><i><font color="Green" title="E101">D3</font></i>: 
( <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/><a NAME="E5:43_17_5_1_1_1_2_2"/><i><font color="Green" title="E102">B</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:43_17_5_1_1_1_2_2"/><i><font color="Green" title="E103">D2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1_2_2"><i><font color="Green" title="E100">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_17_5_1_1_1_2_2"><i><font color="Green" title="E101">D3</font></i></a>, <a class="txt" href="matrix14.html#E5:43_17_5_1_1_1_2_2"><i><font color="Green" title="E102">B</font></i></a>, <a class="txt" href="matrix14.html#E5:43_17_5_1_1_1"><i><font color="Green" title="E94">C2</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_17_5_1_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_17_5_1_1_1_2_2_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_17_5_1_1_1_2_2_1_1_1"/><i><font color="Green" title="E104">E0</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E8:43_17_5_1_1_1"><i><font color="Green" title="E97">C4</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43_17"><i><font color="Green" title="E85">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_1_1_2_2"><i><font color="Green" title="E100">D0</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a>, <a class="txt" href="matrix14.html#E10:43_17_5_1_1_1"><i><font color="Green" title="E99">E1</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><b>then </b><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E7:43_17"><i><font color="Green" title="E85">B4</font></i></a>, <a class="txt" href="matrix14.html#E9:43_17"><i><font color="Green" title="E87">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E9:43_17"><i><font color="Green" title="E87">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1_2_2_1_1_1"><i><font color="Green" title="E104">E0</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:43_17_5_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E6:43_17_5_1_1_1_2_2"><i><font color="Green" title="E103">D2</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1_2_2_1_1_1"><i><font color="Green" title="E104">E0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_17_5_1_1_1_2_2_1_1_2"/><i><font color="Green" title="E104">E0</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E8:43_17_5_1_1_1"><i><font color="Green" title="E97">C4</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43_17"><i><font color="Green" title="E85">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_1_1_2_2"><i><font color="Green" title="E100">D0</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a>, <a class="txt" href="matrix14.html#E10:43_17_5_1_1_1"><i><font color="Green" title="E99">E1</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><b>then </b><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E7:43_17"><i><font color="Green" title="E85">B4</font></i></a>, <a class="txt" href="matrix14.html#E9:43_17"><i><font color="Green" title="E87">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E9:43_17"><i><font color="Green" title="E87">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1_2_2_1_1_2"><i><font color="Green" title="E104">E0</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:43_17_5_1_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E6:43_17_5_1_1_1_2_2"><i><font color="Green" title="E103">D2</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1"><i><font color="Green" title="E91">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_1_1_2_2_1_1_2"><i><font color="Green" title="E104">E0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:43_17_5_1_1_1_2_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</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></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:43_17_5_1_1"/><b>then </b><i><font color="Green" title="E91">B7</font></i>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E6:43_17"><i><font color="Green" title="E84">B3</font></i></a>, <a class="txt" href="matrix14.html#E10:43_17"><i><font color="Green" title="E88">B5</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/><a NAME="E5:43_17_5_1_1"/><i><font color="Green" title="E92">A2a</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><a NAME="E6:43_17_5_1_1"/><i><font color="Green" title="E93">K2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E5:43_17_5_1_1"><i><font color="Green" title="E92">A2a</font></i></a></i>;<br/><i><font color="Green" title="E94">K3</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E6:43_17_5_1_1"><i><font color="Green" title="E93">K2</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E2:43_17"><i><font color="Green" title="E80">B20</font></i></a>, <a class="txt" href="matrix14.html#E5:43_17_5_1_1"><i><font color="Green" title="E92">A2a</font></i></a></i>
;<br/><i><font color="Green" title="E95">K0</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E6:43_17_5_1_1"><i><font color="Green" title="E93">K2</font></i></a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E37:43"><i><font color="Green" title="E67">A10</font></i></a></i>
;<br/><a NAME="E9:43_17_5_1_1"/><i><font color="Green" title="E96">B18</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a></i>;<br/><i><font color="Green" title="E97">B28</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="txt" href="matrix14.html#E5:43_17_5_1_1"><i><font color="Green" title="E92">A2a</font></i></a>, <a class="txt" href="matrix14.html#E28:43"><i><font color="Green" title="E59">A2d</font></i></a>, <a class="txt" href="matrix14.html#E4:43_17"><i><font color="Green" title="E82">A22</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T11" target="_self" title="MATRIX14:th.11">BB248</a>, <a class="txt" href="matrix14.html#E7:43_17"><i><font color="Green" title="E85">B4</font></i></a>, <a class="txt" href="matrix14.html#E6:43_17_5_1_1"><i><font color="Green" title="E93">K2</font></i></a>, <a class="txt" href="matrix14.html#E4:43_17_5_1_1"><i><font color="Green" title="E91">B7</font></i></a>, <a class="txt" href="matrix14.html#E9:43_17"><i><font color="Green" title="E87">B5b</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">BB246</a>, <a class="txt" href="matrix14.html#E6:43_17_5_1_1"><i><font color="Green" title="E93">K2</font></i></a>, <a class="txt" href="matrix14.html#E31:43"><i><font color="Green" title="E62">A2b</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="txt" href="matrix14.html#E8:43_17_5_1_1"><i><font color="Green" title="E95">K0</font></i></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E6:43_17_5_1_1"><i><font color="Green" title="E93">K2</font></i></a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E7:43_17_5_1_1"><i><font color="Green" title="E94">K3</font></i></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a>, <a class="txt" href="matrix14.html#E6:43_17_5_1_1"><i><font color="Green" title="E93">K2</font></i></a></i>
;<br/><b>consider </b><font color="Maroon" title="c10">p1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E12:43_17_5_1_1"/><i><font color="Green" title="E98">A97</font></i>: 
( <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a>, <a class="txt" href="matrix14.html#E3:43_17"><i><font color="Green" title="E81">B2</font></i></a></i>;<br/><a NAME="E13:43_17_5_1_1"/>
<font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E12:43_17_5_1_1"><i><font color="Green" title="E98">A97</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a></i>;<br/><a NAME="E14:43_17_5_1_1"/><b>then </b><i><font color="Green" title="E99">A98</font></i>: 
<font color="Maroon" title="c10">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</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="c2">K</font>
 <b>by </b><i><a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c11">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E16:43_17_5_1_1"/><i><font color="Green" title="E100">A99</font></i>: 
( <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a>, <a class="txt" href="matrix14.html#E5:43_17"><i><font color="Green" title="E83">A23b</font></i></a></i>;<br/><a NAME="E17:43_17_5_1_1"/>
<font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E16:43_17_5_1_1"><i><font color="Green" title="E100">A99</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E10:43_17_5_1_1"><i><font color="Green" title="E97">B28</font></i></a>, <a class="txt" href="matrix14.html#E16:43_17_5_1_1"><i><font color="Green" title="E100">A99</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E9:43_17_5_1_1"><i><font color="Green" title="E96">B18</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E12:43_17_5_1_1"><i><font color="Green" title="E98">A97</font></i></a>, <a class="txt" href="matrix14.html#E14:43_17_5_1_1"><i><font color="Green" title="E99">A98</font></i></a>, <a class="ref" href="rlvect_1.html#T25" title="RLVECT_1:th.25">RLVECT_1:25</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/> <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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_2"><b>suppose </b></a><a NAME="E1:43_17_5_1_2"/><i><font color="Green" title="E90">F1</font></i>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="c2">K</font></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c8">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c9">q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/><a NAME="E3:43_17_5_1_2"/><i><font color="Green" title="E91">F2</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_17_5_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<b>assume </b><a NAME="E1:43_17_5_1_2_1"/><i><font color="Green" title="E92">C0</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<a NAME="E2:43_17_5_1_2_1"/><b>then </b><i><font color="Green" title="E93">C10</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:43_17_5_1_2_1"/><b>then </b><i><font color="Green" title="E94">C10b</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E11:43_17"><i><font color="Green" title="E89">B4b</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:43_17_5_1_2_1"/><i><font color="Green" title="E95">A2b</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E5:43_17_5_1_2_1"/><i><font color="Green" title="E96">C12</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_2_1"><i><font color="Green" title="E92">C0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="matrix14.html#E28:43"><i><font color="Green" title="E59">A2d</font></i></a></i>;<br/>
<a NAME="E6:43_17_5_1_2_1"/><i><font color="Green" title="E97">C2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E5:43_17_5_1_2_1"><i><font color="Green" title="E96">C12</font></i></a></i>;<br/>
<a NAME="E7:43_17_5_1_2_1"/><i><font color="Green" title="E98">A13</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_2_1"><i><font color="Green" title="E92">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E4:43_17_5_1_2_1"><i><font color="Green" title="E95">A2b</font></i></a></i>;<br/>
<a NAME="E8:43_17_5_1_2_1"/><i><font color="Green" title="E99">C4</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a>, <a class="txt" href="matrix14.html#E3:43_17_5_1_2_1"><i><font color="Green" title="E94">C10b</font></i></a>, <a class="txt" href="matrix14.html#E7:43_17_5_1_2_1"><i><font color="Green" title="E98">A13</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_17_5_1_2_1_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> )
 </span><b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_2_1"><i><font color="Green" title="E92">C0</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_17_5_1_2_1_1_1"/><i><font color="Green" title="E100">D0</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E8:43_17_5_1_2_1"><i><font color="Green" title="E99">C4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_2_1_1_1"><i><font color="Green" title="E100">D0</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_2_1"><i><font color="Green" title="E92">C0</font></i></a>, <a class="txt" href="matrix14.html#E7:43_17_5_1_2_1"><i><font color="Green" title="E98">A13</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<b>by </b><i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
;<br/><b>hence </b><a NAME="E3:43_17_5_1_2_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_2"><i><font color="Green" title="E90">F1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="txt" href="matrix14.html#E6:43_17_5_1_2_1"><i><font color="Green" title="E97">C2</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_2_1_1_1"><i><font color="Green" title="E100">D0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_17_5_1_2_1_1_2"/><i><font color="Green" title="E100">D0</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_17_5_1_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E42:43"><i><font color="Green" title="E72">A8d</font></i></a>, <a class="txt" href="matrix14.html#E2:43_17"><i><font color="Green" title="E80">B20</font></i></a>, <a class="txt" href="matrix14.html#E2:43_17_5_1_2_1"><i><font color="Green" title="E93">C10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E4:43_17_5_1_2_1_1_2"/><i><font color="Green" title="E101">D3</font></i>: 
( <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/><a NAME="E5:43_17_5_1_2_1_1_2"/><i><font color="Green" title="E102">B</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:43_17_5_1_2_1_1_2"/><i><font color="Green" title="E103">D2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_2_1_1_2"><i><font color="Green" title="E100">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_2_1"><i><font color="Green" title="E92">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_17_5_1_2_1_1_2"><i><font color="Green" title="E101">D3</font></i></a>, <a class="txt" href="matrix14.html#E5:43_17_5_1_2_1_1_2"><i><font color="Green" title="E102">B</font></i></a>, <a class="txt" href="matrix14.html#E6:43_17_5_1_2_1"><i><font color="Green" title="E97">C2</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_17_5_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_17_5_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_17_5_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E104">E0</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_17_5_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E105">A13</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_2_1"><i><font color="Green" title="E92">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E4:43_17_5_1_2_1"><i><font color="Green" title="E95">A2b</font></i></a></i>;<br/><a NAME="E3:43_17_5_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E106">A13a</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_2_1"><i><font color="Green" title="E92">C0</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_17_5_1_2_1_1_2_1_1_1"><i><font color="Green" title="E105">A13</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_2_1_1_2"><i><font color="Green" title="E100">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_2_1"><i><font color="Green" title="E92">C0</font></i></a>, <a class="txt" href="matrix14.html#E3:43_17_5_1_2_1_1_2_1_1_1"><i><font color="Green" title="E106">A13a</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><b>hence </b><a NAME="E5:43_17_5_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_2_1_1_2_1_1_1"><i><font color="Green" title="E104">E0</font></i></a>, <a class="txt" href="matrix14.html#E6:43_17_5_1_2_1_1_2"><i><font color="Green" title="E103">D2</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_2_1"><i><font color="Green" title="E92">C0</font></i></a>, <a class="txt" href="matrix14.html#E8:43_17_5_1_2_1"><i><font color="Green" title="E99">C4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_17_5_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_17_5_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><b>hence </b><a NAME="E2:43_17_5_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_17_5_1_2"><i><font color="Green" title="E90">F1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17_5_1_2_1_1_2"><i><font color="Green" title="E100">D0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:43_17_5_1_2_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</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></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E4:43_17_5_1_2"/><i><font color="Green" title="E92">A2a</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/><a NAME="E5:43_17_5_1_2"/><i><font color="Green" title="E93">K2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E4:43_17_5_1_2"><i><font color="Green" title="E92">A2a</font></i></a></i>;<br/><i><font color="Green" title="E94">K0</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E5:43_17_5_1_2"><i><font color="Green" title="E93">K2</font></i></a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E37:43"><i><font color="Green" title="E67">A10</font></i></a></i>
;<br/><a NAME="E7:43_17_5_1_2"/><i><font color="Green" title="E95">B18</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a></i>;<br/><i><font color="Green" title="E96">B28</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="txt" href="matrix14.html#E4:43_17_5_1_2"><i><font color="Green" title="E92">A2a</font></i></a>, <a class="txt" href="matrix14.html#E28:43"><i><font color="Green" title="E59">A2d</font></i></a>, <a class="txt" href="matrix14.html#E4:43_17"><i><font color="Green" title="E82">A22</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="txt" href="matrix14.html#E3:43_17_5_1_2"><i><font color="Green" title="E91">F2</font></i></a>, <a class="txt" href="matrix14.html#E6:43_17"><i><font color="Green" title="E84">B3</font></i></a>, <a class="txt" href="matrix14.html#E11:43_17"><i><font color="Green" title="E89">B4b</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">BB246</a>, <a class="txt" href="matrix14.html#E5:43_17_5_1_2"><i><font color="Green" title="E93">K2</font></i></a>, <a class="txt" href="matrix14.html#E31:43"><i><font color="Green" title="E62">A2b</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E6:43_17_5_1_2"><i><font color="Green" title="E94">K0</font></i></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E5:43_17_5_1_2"><i><font color="Green" title="E93">K2</font></i></a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a></i>
;<br/><b>consider </b><font color="Maroon" title="c10">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E10:43_17_5_1_2"/><i><font color="Green" title="E97">A99</font></i>: 
( <font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a>, <a class="txt" href="matrix14.html#E5:43_17"><i><font color="Green" title="E83">A23b</font></i></a></i>;<br/><a NAME="E11:43_17_5_1_2"/>
<font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E10:43_17_5_1_2"><i><font color="Green" title="E97">A99</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E8:43_17_5_1_2"><i><font color="Green" title="E96">B28</font></i></a>, <a class="txt" href="matrix14.html#E10:43_17_5_1_2"><i><font color="Green" title="E97">A99</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E7:43_17_5_1_2"><i><font color="Green" title="E95">B18</font></i></a>, <a class="txt" href="matrix14.html#E1:43_17"><i><font color="Green" title="E79">B1</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/> <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><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:43_17"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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="c2">K</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>
<a NAME="E49:43"/><i><font color="Green" title="E79">A85</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font> )</span><br/>



<b>assume </b><a NAME="E1:43_18"/><i><font color="Green" title="E80">B1</font></i>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span><br/>

<a NAME="E2:43_18"/><b>then </b><i><font color="Green" title="E81">B20</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:43_18"/><i><font color="Green" title="E82">B2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></i>;<br/>
<a NAME="E4:43_18"/><i><font color="Green" title="E83">A22</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></i>;<br/>
<a NAME="E5:43_18"/><i><font color="Green" title="E84">A23b</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E2:43"><i><font color="Green" title="E35">A40</font></i></a></i>;<br/>
<i><font color="Green" title="E85">B3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>
;<br/>
<i><font color="Green" title="E86">B4</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<i><font color="Green" title="E87">B4d</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<a NAME="E9:43_18"/><i><font color="Green" title="E88">B5b</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E10:43_18"/><b>then </b><i><font color="Green" title="E89">B5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E7:43_18"><i><font color="Green" title="E86">B4</font></i></a>, <a class="ref" href="matrix14.html#T2" target="_self" title="MATRIX14:th.2">Th6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c9">q</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/>
<a NAME="E12:43_18"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_18_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<b>assume </b><a NAME="E1:43_18_4"/><i><font color="Green" title="E90">C0</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<a NAME="E2:43_18_4"/><b>then </b>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:43_18_4"/><b>then </b><i><font color="Green" title="E91">C10b</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E8:43_18"><i><font color="Green" title="E87">B4d</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green" title="E92">C10a</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<b>by </b><i><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></i>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>
;<br/>
<a NAME="E5:43_18_4"/><i><font color="Green" title="E93">C12</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="matrix14.html#E28:43"><i><font color="Green" title="E59">A2d</font></i></a></i>;<br/>
<a NAME="E6:43_18_4"/><i><font color="Green" title="E94">C2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a>, <a class="txt" href="matrix14.html#E5:43_18_4"><i><font color="Green" title="E93">C12</font></i></a></i>;<br/>
<a NAME="E7:43_18_4"/><i><font color="Green" title="E95">A2b</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/>
<a NAME="E8:43_18_4"/><i><font color="Green" title="E96">A13</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43_18_4"><i><font color="Green" title="E95">A2b</font></i></a></i>;<br/>
<i><font color="Green" title="E97">C4</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<b>by </b><i><a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a>, <a class="txt" href="matrix14.html#E3:43_18_4"><i><font color="Green" title="E91">C10b</font></i></a>, <a class="txt" href="matrix14.html#E8:43_18_4"><i><font color="Green" title="E96">A13</font></i></a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_18_4_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_18_4_3"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> )
 </span><b>by </b><i><a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_18_4_3_1"><b>suppose </b></a><a NAME="E1:43_18_4_3_1"/><i><font color="Green" title="E98">D0</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_18_4_3_1"/><i><font color="Green" title="E99">D2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E3:43"><i><font color="Green" title="E36">A1b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a>, <a class="txt" href="matrix14.html#E6:43_18_4"><i><font color="Green" title="E94">C2</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4_3_1"><i><font color="Green" title="E98">D0</font></i></a></i>;<br/><a NAME="E3:43_18_4_3_1"/><i><font color="Green" title="E100">K3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_18_4"><i><font color="Green" title="E92">C10a</font></i></a>, <a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>;<br/><i><font color="Green" title="E101">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E9:43_18"><i><font color="Green" title="E88">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_18_4_3_1"><i><font color="Green" title="E98">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a></i>
;<br/><i><font color="Green" title="E102">D1</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E9:43_18_4"><i><font color="Green" title="E97">C4</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E3:43_18_4_3_1"><i><font color="Green" title="E100">K3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_18_4_3_1"><i><font color="Green" title="E98">D0</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="matrix14.html#E8:43_18_4"><i><font color="Green" title="E96">A13</font></i></a></i>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
;<br/><a NAME="E6:43_18_4_3_1"/>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E5:43_18_4_3_1"><i><font color="Green" title="E102">D1</font></i></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E7:43_18"><i><font color="Green" title="E86">B4</font></i></a>, <a class="txt" href="matrix14.html#E9:43_18"><i><font color="Green" title="E88">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_18_4_3_1"><i><font color="Green" title="E101">K1</font></i></a></i>;<br/><b>hence </b><a NAME="E7:43_18_4_3_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E2:43_18_4_3_1"><i><font color="Green" title="E99">D2</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_18_4_3_2"><b>suppose </b></a><a NAME="E1:43_18_4_3_2"/><i><font color="Green" title="E98">D0</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_18_4_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E4:43_18_4_3_2"/><i><font color="Green" title="E99">D3</font></i>: 
( <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></i>;<br/><a NAME="E5:43_18_4_3_2"/><i><font color="Green" title="E100">B</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E6:43_18_4_3_2"/><i><font color="Green" title="E101">D2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4_3_2"><i><font color="Green" title="E98">D0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_18_4_3_2"><i><font color="Green" title="E99">D3</font></i></a>, <a class="txt" href="matrix14.html#E5:43_18_4_3_2"><i><font color="Green" title="E100">B</font></i></a>, <a class="txt" href="matrix14.html#E6:43_18_4"><i><font color="Green" title="E94">C2</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_18_4_3_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_18_4_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_18_4_3_2_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_18_4_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_18_4_3_2_1_1_1"/><i><font color="Green" title="E102">E0</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_18_4_3_2_1_1_1"/><b>then </b><i><font color="Green" title="E103">E2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</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="c2">K</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E6:43_18_4_3_2"><i><font color="Green" title="E101">D2</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a></i>;<br/><a NAME="E3:43_18_4_3_2_1_1_1"/><i><font color="Green" title="E104">A3a</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/><a NAME="E4:43_18_4_3_2_1_1_1"/><i><font color="Green" title="E105">E1</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E3:43_18_4_3_2_1_1_1"><i><font color="Green" title="E104">A3a</font></i></a></i>;<br/><i><font color="Green" title="E106">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E9:43_18"><i><font color="Green" title="E88">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4_3_2_1_1_1"><i><font color="Green" title="E102">E0</font></i></a></i>
;<br/><a NAME="E6:43_18_4_3_2_1_1_1"/><i><font color="Green" title="E107">K2</font></i>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43_18"><i><font color="Green" title="E86">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a></i>;<br/><i><font color="Green" title="E108">E3</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_18_4_3_2"><i><font color="Green" title="E98">D0</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_18_4_3_2_1_1_1"><i><font color="Green" title="E105">E1</font></i></a>, <a class="txt" href="matrix14.html#E9:43_18_4"><i><font color="Green" title="E97">C4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><a NAME="E8:43_18_4_3_2_1_1_1"/>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix14.html#E7:43_18_4_3_2_1_1_1"><i><font color="Green" title="E108">E3</font></i></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E7:43_18"><i><font color="Green" title="E86">B4</font></i></a>, <a class="txt" href="matrix14.html#E9:43_18"><i><font color="Green" title="E88">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="txt" href="matrix14.html#E5:43_18_4_3_2_1_1_1"><i><font color="Green" title="E106">K1</font></i></a>, <a class="txt" href="matrix14.html#E6:43_18_4_3_2_1_1_1"><i><font color="Green" title="E107">K2</font></i></a></i>;<br/><b>hence </b><a NAME="E9:43_18_4_3_2_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E2:43_18_4_3_2_1_1_1"><i><font color="Green" title="E103">E2</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_18_4_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_18_4_3_2_1_1_2"/><i><font color="Green" title="E102">E0</font></i>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_18_4_3_2_1_1_2"/><b>then </b><i><font color="Green" title="E103">E2</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E6:43_18_4_3_2"><i><font color="Green" title="E101">D2</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a></i>;<br/><a NAME="E3:43_18_4_3_2_1_1_2"/><i><font color="Green" title="E104">A3a</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">AA1100</a></i>;<br/><a NAME="E4:43_18_4_3_2_1_1_2"/><i><font color="Green" title="E105">E1</font></i>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E3:43_18_4_3_2_1_1_2"><i><font color="Green" title="E104">A3a</font></i></a></i>;<br/><i><font color="Green" title="E106">K1</font></i>: <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E9:43_18"><i><font color="Green" title="E88">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4_3_2_1_1_2"><i><font color="Green" title="E102">E0</font></i></a></i>
;<br/><a NAME="E6:43_18_4_3_2_1_1_2"/><i><font color="Green" title="E107">K2</font></i>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E7:43_18"><i><font color="Green" title="E86">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a></i>;<br/><i><font color="Green" title="E108">E3</font></i>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E1:43_18_4_3_2"><i><font color="Green" title="E98">D0</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_18_4_3_2_1_1_2"><i><font color="Green" title="E105">E1</font></i></a>, <a class="txt" href="matrix14.html#E9:43_18_4"><i><font color="Green" title="E97">C4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><a NAME="E8:43_18_4_3_2_1_1_2"/>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">BB120</a>, <a class="txt" href="matrix14.html#E7:43_18"><i><font color="Green" title="E86">B4</font></i></a>, <a class="txt" href="matrix14.html#E9:43_18"><i><font color="Green" title="E88">B5b</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18_4"><i><font color="Green" title="E90">C0</font></i></a>, <a class="txt" href="matrix14.html#E7:43_18_4_3_2_1_1_2"><i><font color="Green" title="E108">E3</font></i></a>, <a class="txt" href="matrix14.html#E5:43_18_4_3_2_1_1_2"><i><font color="Green" title="E106">K1</font></i></a>, <a class="txt" href="matrix14.html#E6:43_18_4_3_2_1_1_2"><i><font color="Green" title="E107">K2</font></i></a></i>;<br/><b>hence </b><a NAME="E9:43_18_4_3_2_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E2:43_18_4_3_2_1_1_2"><i><font color="Green" title="E103">E2</font></i></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:43_18_4_3_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</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></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E13:43_18"/><b>then </b><i><font color="Green" title="E90">B7</font></i>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix14.html#E6:43_18"><i><font color="Green" title="E85">B3</font></i></a>, <a class="txt" href="matrix14.html#E10:43_18"><i><font color="Green" title="E89">B5</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>
<a NAME="E14:43_18"/><i><font color="Green" title="E91">K2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E29:43"><i><font color="Green" title="E60">A2a</font></i></a></i>;<br/>
<a NAME="E15:43_18"/><i><font color="Green" title="E92">A2a</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></i>;<br/>
<i><font color="Green" title="E93">K3</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E14:43_18"><i><font color="Green" title="E91">K2</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E2:43_18"><i><font color="Green" title="E81">B20</font></i></a>, <a class="txt" href="matrix14.html#E15:43_18"><i><font color="Green" title="E92">A2a</font></i></a></i>
;<br/>
<i><font color="Green" title="E94">K0</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a>, <a class="txt" href="matrix14.html#E14:43_18"><i><font color="Green" title="E91">K2</font></i></a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrix14.html#E37:43"><i><font color="Green" title="E67">A10</font></i></a></i>
;<br/>
<a NAME="E18:43_18"/><i><font color="Green" title="E95">B18</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a></i>;<br/>
<i><font color="Green" title="E96">B28</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a>, <a class="txt" href="matrix14.html#E15:43_18"><i><font color="Green" title="E92">A2a</font></i></a>, <a class="txt" href="matrix14.html#E28:43"><i><font color="Green" title="E59">A2d</font></i></a>, <a class="txt" href="matrix14.html#E4:43_18"><i><font color="Green" title="E83">A22</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T11" target="_self" title="MATRIX14:th.11">BB248</a>, <a class="txt" href="matrix14.html#E7:43_18"><i><font color="Green" title="E86">B4</font></i></a>, <a class="txt" href="matrix14.html#E14:43_18"><i><font color="Green" title="E91">K2</font></i></a>, <a class="txt" href="matrix14.html#E13:43_18"><i><font color="Green" title="E90">B7</font></i></a>, <a class="txt" href="matrix14.html#E9:43_18"><i><font color="Green" title="E88">B5b</font></i></a></i>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">BB200</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">BB246</a>, <a class="txt" href="matrix14.html#E14:43_18"><i><font color="Green" title="E91">K2</font></i></a>, <a class="txt" href="matrix14.html#E31:43"><i><font color="Green" title="E62">A2b</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="txt" href="matrix14.html#E17:43_18"><i><font color="Green" title="E94">K0</font></i></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E14:43_18"><i><font color="Green" title="E91">K2</font></i></a>, <a class="txt" href="matrix14.html#E36:43"><i><font color="Green" title="E66">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E16:43_18"><i><font color="Green" title="E93">K3</font></i></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">AA2627</a>, <a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a>, <a class="txt" href="matrix14.html#E14:43_18"><i><font color="Green" title="E91">K2</font></i></a></i>
;<br/>
<b>consider </b><font color="Maroon" title="c10">p1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E21:43_18"/><i><font color="Green" title="E97">A97</font></i>: 
( <font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a>, <a class="txt" href="matrix14.html#E3:43_18"><i><font color="Green" title="E82">B2</font></i></a></i>;<br/>
<a NAME="E22:43_18"/>
<font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E21:43_18"><i><font color="Green" title="E97">A97</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a></i>;<br/>
<a NAME="E23:43_18"/><b>then </b><i><font color="Green" title="E98">A98</font></i>: 
<font color="Maroon" title="c10">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 
<b>by </b><i><a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><b> such that </b><br/><a NAME="E25:43_18"/><i><font color="Green" title="E99">A99</font></i>: 
( <font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a>, <a class="txt" href="matrix14.html#E5:43_18"><i><font color="Green" title="E84">A23b</font></i></a></i>;<br/>
<a NAME="E26:43_18"/>
<font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="txt" href="matrix14.html#E1:43"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="matrix14.html#E25:43_18"><i><font color="Green" title="E99">A99</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E19:43_18"><i><font color="Green" title="E96">B28</font></i></a>, <a class="txt" href="matrix14.html#E25:43_18"><i><font color="Green" title="E99">A99</font></i></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E18:43_18"><i><font color="Green" title="E95">B18</font></i></a>, <a class="txt" href="matrix14.html#E1:43_18"><i><font color="Green" title="E80">B1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix14.html#E21:43_18"><i><font color="Green" title="E97">A97</font></i></a>, <a class="txt" href="matrix14.html#E23:43_18"><i><font color="Green" title="E98">A98</font></i></a>, <a class="ref" href="rlvect_1.html#T25" title="RLVECT_1:th.25">RLVECT_1:25</a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
;<br/> <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>
<a NAME="E50:43"/><i><font color="Green" title="E80">KK1</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> )</span><br/>

<b>assume </b><a NAME="E1:43_19"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/>

<a NAME="E2:43_19"/><b>then </b><i><font color="Green" title="E81">B2</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="matrix14.html#E45:43"><i><font color="Green" title="E75">A18</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:43_19"/><b>then </b><i><font color="Green" title="E82">B3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:43_19"/><i><font color="Green" title="E83">B4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_19"><i><font color="Green" title="E81">B2</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">i0</font> = <font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c9">j0</font> = <font color="Maroon" title="c7">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_19_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> or 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> )
 </span><b>by </b><i><a class="txt" href="matrix14.html#E3:43_19"><i><font color="Green" title="E82">B3</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_1"><b>suppose </b></a><a NAME="E1:43_19_1_1"/><i><font color="Green" title="E84">C0</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_19_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_19_1_1_1_1"/>
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_19_1_1_1_1_1"/><i><font color="Green" title="E85">D0</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><i><font color="Green" title="E86">D1</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<b>by </b><i><a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">BB300</a>, <a class="txt" href="matrix14.html#E3:43_19"><i><font color="Green" title="E82">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_19"><i><font color="Green" title="E83">B4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E1:43_19_1_1_1_1_1"><i><font color="Green" title="E85">D0</font></i></a>, <a class="txt" href="matrix14.html#E3:43_19"><i><font color="Green" title="E82">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_19"><i><font color="Green" title="E83">B4</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i0</font>,<font color="Maroon" title="c9">j0</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_19_1_1_1_1_1"><i><font color="Green" title="E86">D1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_19_1_1"><i><font color="Green" title="E84">C0</font></i></a>, <a class="txt" href="matrix14.html#E3:43_19"><i><font color="Green" title="E82">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_19"><i><font color="Green" title="E83">B4</font></i></a>, <a class="txt" href="matrix14.html#E48:43"><i><font color="Green" title="E78">A15</font></i></a>, <a class="txt" href="matrix14.html#E1:43_19_1_1_1_1_1"><i><font color="Green" title="E85">D0</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_19_1_1_1_1_2"/><i><font color="Green" title="E85">D0</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><i><font color="Green" title="E86">D1</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<b>by </b><i><a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">BB300</a>, <a class="txt" href="matrix14.html#E3:43_19"><i><font color="Green" title="E82">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_19"><i><font color="Green" title="E83">B4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E1:43_19_1_1_1_1_2"><i><font color="Green" title="E85">D0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_19"><i><font color="Green" title="E83">B4</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i0</font>,<font color="Maroon" title="c9">j0</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_19_1_1_1_1_2"><i><font color="Green" title="E86">D1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_19_1_1"><i><font color="Green" title="E84">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_19"><i><font color="Green" title="E83">B4</font></i></a>, <a class="txt" href="matrix14.html#E49:43"><i><font color="Green" title="E79">A85</font></i></a>, <a class="txt" href="matrix14.html#E1:43_19_1_1_1_1_2"><i><font color="Green" title="E85">D0</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:43_19_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_2"><b>suppose </b></a><a NAME="E1:43_19_1_2"/><i><font color="Green" title="E84">C0</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_19_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_19_1_2_1_1"/>
( <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_19_1_2_1_1_1"/><i><font color="Green" title="E85">D0</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><i><font color="Green" title="E86">D1</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<b>by </b><i><a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">BB300</a>, <a class="txt" href="matrix14.html#E3:43_19"><i><font color="Green" title="E82">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_19"><i><font color="Green" title="E83">B4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">AA1120</a>, <a class="txt" href="matrix14.html#E3:43_19"><i><font color="Green" title="E82">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_19"><i><font color="Green" title="E83">B4</font></i></a>, <a class="txt" href="matrix14.html#E1:43_19_1_2_1_1_1"><i><font color="Green" title="E85">D0</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i0</font>,<font color="Maroon" title="c9">j0</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_19_1_2_1_1_1"><i><font color="Green" title="E86">D1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_19_1_2"><i><font color="Green" title="E84">C0</font></i></a>, <a class="txt" href="matrix14.html#E4:43_19"><i><font color="Green" title="E83">B4</font></i></a>, <a class="txt" href="matrix14.html#E47:43"><i><font color="Green" title="E77">A14</font></i></a>, <a class="txt" href="matrix14.html#E1:43_19_1_2_1_1_1"><i><font color="Green" title="E85">D0</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_19_1_2_1_1_2"/><i><font color="Green" title="E85">D9</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><a NAME="E2:43_19_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E86">D0</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E4:43_19"><i><font color="Green" title="E83">B4</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a>, <a class="txt" href="matrix14.html#E1:43_19_1_2"><i><font color="Green" title="E84">C0</font></i></a></i>;<br/><i><font color="Green" title="E87">D1</font></i>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<b>by </b><i><a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">BB300</a>, <a class="txt" href="matrix14.html#E3:43_19"><i><font color="Green" title="E82">B3</font></i></a>, <a class="txt" href="matrix14.html#E4:43_19"><i><font color="Green" title="E83">B4</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<b>by </b><i><a class="txt" href="matrix14.html#E2:43_19_1_2_1_1_2"><i><font color="Green" title="E86">D0</font></i></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">AA1110</a>, <a class="txt" href="matrix14.html#E4:43_19"><i><font color="Green" title="E83">B4</font></i></a></i>
;<br/><b>thus </b><a NAME="E4:43_19_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
 <b>by </b><i><a class="txt" href="matrix14.html#E3:43_19_1_2_1_1_2"><i><font color="Green" title="E87">D1</font></i></a>, <a class="txt" href="matrix14.html#E1:43_19_1_2"><i><font color="Green" title="E84">C0</font></i></a>, <a class="txt" href="matrix14.html#E46:43"><i><font color="Green" title="E76">A13</font></i></a>, <a class="txt" href="matrix14.html#E1:43_19_1_2_1_1_2"><i><font color="Green" title="E85">D9</font></i></a>, <a class="txt" href="matrix14.html#E4:43_19"><i><font color="Green" title="E83">B4</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:43_19_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</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></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E51:43"/><i><font color="Green" title="E81">A25</font></i>: 
<font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a>, <a class="txt" href="matrix14.html#E50:43"><i><font color="Green" title="E80">KK1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E52:43"/>
<font color="Maroon" title="c4">P</font> is <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>
 <b>by </b><i><a class="ref" href="matrix14.html#T19" target="_self" title="MATRIX14:th.19">AA4145</a>, <a class="txt" href="matrix14.html#E27:43"><i><font color="Green" title="E58">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span><br/>

<b>thus </b><a NAME="E53:43"/>
<font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> 
 <b>by </b><i><a class="ref" href="matrix14.html#T18" target="_self" title="MATRIX14:th.18">AA4140</a>, <a class="txt" href="matrix14.html#E27:43"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="matrix14.html#E51:43"><i><font color="Green" title="E81">A25</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
