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<span class="kw">consider </span><font color="Maroon" title="c7">I</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="prvect_4.html#K3" title="PRVECT_4:func.3">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K3" title="PRVECT_4:func.3">:]</a></span>,<span class="p1">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:27_1_1"/>
( <font color="Maroon" title="c7">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c7">I</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )
 <span class="kw">and </span><br/><a NAME="E3:27_1_1"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">E</font><br/>  for <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">G</font> holds  <font color="Maroon" title="c7">I</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E4:27_1_1"/>
( (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K3" title="PRVECT_4:func.3">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K3" title="PRVECT_4:func.3">:]</a></span> holds  <font color="Maroon" title="c7">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">w</font></span>)</span> ) &amp; (  for <font color="Olive" title="b1">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_4.html#K3" title="PRVECT_4:func.3">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K3" title="PRVECT_4:func.3">:]</a></span><br/>  for <font color="Olive" title="b2">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c7">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> ) &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2"><a href="prvect_4.html#K3" title="PRVECT_4:func.3">[:</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="prvect_4.html#K3" title="PRVECT_4:func.3">:]</a></span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_4.html#T7" target="_self" onmouseover="rs('prvect_4/T7')" onmouseout="rh()">Th11</a></span>;<br/>
<a NAME="E5:27_1_1"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c7">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="prvect_4.html#K4" title="PRVECT_4:func.4">[</a><span class="default"><font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="prvect_4.html#K4" title="PRVECT_4:func.4">]</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T112" onmouseover="rs('funct_2/T112')" onmouseout="rh()">FUNCT_2:112</a></span>;<br/>
<a NAME="E6:27_1_1"/>
<font color="Maroon" title="c7">I</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> (<font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:27_1_1"/>
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K6" title="PRVECT_2:func.6">product</a> <span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1_1"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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