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<span class="kw">let </span><font color="Maroon" title="c1">F1</font>, <font color="Maroon" title="c2">F2</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <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"> ( <font color="Maroon" title="c1">F1</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c2">F2</font> &amp; <font color="Maroon" title="c2">F2</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c1">F1</font> implies  for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">F1</font>,<font color="Maroon" title="c2">F2</font>  st <font color="Olive" title="b1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">F1</font> holds <br/><font color="Olive" title="b1">f</font> is  <a href="mod_4.html#V7" title="MOD_4:attr.7">isomorphism</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:153"/><span class="lab"><font color="Green" title="E69">AS1</font></span>: 
( <font color="Maroon" title="c1">F1</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c2">F2</font> &amp; <font color="Maroon" title="c2">F2</font> is    <a href="ec_pf_1.html#M1" title="EC_PF_1:mode.1">Subfield</a> of <font color="Maroon" title="c1">F1</font> )
 ; <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">  for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">F1</font>,<font color="Maroon" title="c2">F2</font>  st <font color="Olive" title="b1">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">F1</font> holds <br/><font color="Olive" title="b1">f</font> is  <a href="mod_4.html#V7" title="MOD_4:attr.7">isomorphism</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">F1</font>,<font color="Maroon" title="c2">F2</font>; <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"> ( <font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">F1</font> implies <font color="Maroon" title="c3">f</font> is  <a href="mod_4.html#V7" title="MOD_4:attr.7">isomorphism</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:153"/><span class="lab"><font color="Green" title="E70">AS2</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">F1</font>
 ; <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"> <font color="Maroon" title="c3">f</font> is  <a href="mod_4.html#V7" title="MOD_4:attr.7">isomorphism</a> </span><br/>

<a NAME="E3:153"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F1</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="c2">F2</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">F2</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">F1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:153"><span class="lab"><font color="Green" title="E69">AS1</font></span></a>, <a class="ref" href="ec_pf_1.html#D1" onmouseover="rs('ec_pf_1/D1')" onmouseout="rh()">EC_PF_1:def 1</a></span>;<br/>
<a NAME="E4:153"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>;<br/>
<span class="lab"><font color="Green" title="E72">C</font></span>:  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">F1</font> = 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">F2</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:153"><span class="lab"><font color="Green" title="E69">AS1</font></span></a>, <a class="ref" href="ec_pf_1.html#D1" onmouseover="rs('ec_pf_1/D1')" onmouseout="rh()">EC_PF_1:def 1</a></span>
<br/>.= 
 the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">F2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:153"><span class="lab"><font color="Green" title="E71">A</font></span></a></span>
;<br/>
<span class="lab"><font color="Green" title="E73">D</font></span>:  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">F1</font> = 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">F2</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:153"><span class="lab"><font color="Green" title="E69">AS1</font></span></a>, <a class="ref" href="ec_pf_1.html#D1" onmouseover="rs('ec_pf_1/D1')" onmouseout="rh()">EC_PF_1:def 1</a></span>
<br/>.= 
 the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">F2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:153"><span class="lab"><font color="Green" title="E71">A</font></span></a></span>
;<br/>
<a NAME="E7:153"/>
( <font color="Maroon" title="c3">f</font> is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Maroon" title="c3">f</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:153"><span class="lab"><font color="Green" title="E72">C</font></span></a>, <a class="txt" href="#E6:153"><span class="lab"><font color="Green" title="E73">D</font></span></a>, <a class="txt" href="#E2:153"><span class="lab"><font color="Green" title="E70">AS2</font></span></a>, <a class="txt" href="#E1:153"><span class="lab"><font color="Green" title="E69">AS1</font></span></a>, <a class="ref" href="ec_pf_1.html#D1" onmouseover="rs('ec_pf_1/D1')" onmouseout="rh()">EC_PF_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:153"/>
<font color="Maroon" title="c3">f</font> is  <a href="mod_4.html#V7" title="MOD_4:attr.7">isomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:153"><span class="lab"><font color="Green" title="E71">A</font></span></a>, <a class="txt" href="#E2:153"><span class="lab"><font color="Green" title="E70">AS2</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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