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<span class="kw">set </span><font color="Maroon" title="c2">KK</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrix11.html#K5" title="MATRIX11:func.5">addFinS</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K5" title="MATRIX11:func.5">addFinS</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font></span><br/>

<a NAME="E1:75_1_1"/><span class="lab"><font color="Green" title="E64">A1</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E2:75_1_1"/><span class="lab"><font color="Green" title="E65">A2</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="matrix11.html#E68"><span class="lab"><font color="Green" title="E63">Lm9</font></span></a></span>;<br/>
<a NAME="E3:75_1_1"/><span class="lab"><font color="Green" title="E66">A3</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="matrix11.html#E68"><span class="lab"><font color="Green" title="E63">Lm9</font></span></a></span>;<br/>
<a NAME="E4:75_1_1"/><span class="lab"><font color="Green" title="E67">A4</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">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">p1</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:75_1_1_1"/><span class="lab"><font color="Green" title="E68">A5</font></span>: 
<font color="Maroon" title="c5">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">p1</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font></span><br/><a NAME="E2:75_1_1_1"/>
<font color="Maroon" title="c5">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">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="txt" href="#E1:75_1_1_1"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E3:75_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A6</font></span>: 
<font color="Maroon" title="c4">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E4:75_1_1_1"/>
<font color="Maroon" title="c5">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="c3">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="txt" href="#E1:75_1_1_1"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:75_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">p1k</font> = <font color="Maroon" title="c3">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>, <font color="Maroon" title="c7">p2k</font> = <font color="Maroon" title="c4">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75_1_1"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E1:75_1_1"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E3:75_1_1_1"><span class="lab"><font color="Green" title="E69">A6</font></span></a></span>;<br/><a NAME="E7:75_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1k</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">p2k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_1"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="fvsum_1.html#T21" title="FVSUM_1:th.21">FVSUM_1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E8:75_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c4">p2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p2</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c3">p1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75_1_1"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="txt" href="#E3:75_1_1"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="txt" href="#E1:75_1_1_1"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="fvsum_1.html#T21" title="FVSUM_1:th.21">FVSUM_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:75_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A7</font></span>: 
<font color="Maroon" title="c3">p1</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c3">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:75_1_1"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="txt" href="matrix11.html#E68"><span class="lab"><font color="Green" title="E63">Lm9</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/>
<a NAME="E7:75_1_1"/>
<span class="p1">(<span class="default"><a href="matrix11.html#K5" title="MATRIX11:func.5">addFinS</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <font color="Maroon" title="c4">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix11.html#D5" target="_self" title="MATRIX11:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:75_1_1"/>
<span class="p1">(<span class="default"><a href="matrix11.html#K5" title="MATRIX11:func.5">addFinS</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix11.html#K5" title="MATRIX11:func.5">addFinS</a> <font color="Maroon" title="c1">K</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:75_1_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="ref" href="matrix11.html#D5" target="_self" title="MATRIX11:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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