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<b>let </b><font color="Maroon" title="c1">I</font> be  <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">F'</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a> <a href="vectsp_1.html#NV11" title="VECTSP_1:NV.11">Field-like</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">F</font>;<br/><b>let </b><font color="Maroon" title="c5">f'</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">F'</font>;<br/>







<b>assume </b><a NAME="E1:103"/><i><font color="Green" title="E78">A1</font></i>: 
( <font color="Maroon" title="c1">I</font> <a href="quofield.html#R3" title="QUOFIELD:pred.3">has_Field_of_Quotients_Pair</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c1">I</font> <a href="quofield.html#R3" title="QUOFIELD:pred.3">has_Field_of_Quotients_Pair</a> <font color="Maroon" title="c3">F'</font>,<font color="Maroon" title="c5">f'</font> )
 ;<br/>

<a NAME="E2:103"/><b>then </b><i><font color="Green" title="E79">A2</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a>
 
<b>by </b><i><a class="ref" href="quofield.html#D29" target="_self" title="QUOFIELD:def.29">Def29</a></i>;<br/>
<a NAME="E3:103"/><i><font color="Green" title="E80">A3</font></i>: 
<font color="Maroon" title="c5">f'</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E1:103"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="quofield.html#D29" target="_self" title="QUOFIELD:def.29">Def29</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">h1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">F'</font><b> such that </b><br/><a NAME="E5:103"/><i><font color="Green" title="E81">A4</font></i>: 
( <font color="Maroon" title="c6">h1</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a> &amp; <font color="Maroon" title="c6">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f'</font> &amp; ( for <font color="Olive" title="b1">h'</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">F'</font>  st <font color="Olive" title="b1">h'</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a> &amp; <font color="Olive" title="b1">h'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f'</font> holds <br/><font color="Olive" title="b1">h'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">h1</font> ) )
 <b>by </b><i><a class="txt" href="quofield.html#E1:103"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="quofield.html#D29" target="_self" title="QUOFIELD:def.29">Def29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">h2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">F'</font>,<font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E7:103"/><i><font color="Green" title="E82">A5</font></i>: 
( <font color="Maroon" title="c7">h2</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a> &amp; <font color="Maroon" title="c7">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">f</font> &amp; ( for <font color="Olive" title="b1">h'</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">F'</font>,<font color="Maroon" title="c2">F</font>  st <font color="Olive" title="b1">h'</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a> &amp; <font color="Olive" title="b1">h'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">f</font> holds <br/><font color="Olive" title="b1">h'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">h2</font> ) )
 <b>by </b><i><a class="txt" href="quofield.html#E1:103"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="quofield.html#E2:103"><i><font color="Green" title="E79">A2</font></i></a>, <a class="ref" href="quofield.html#D29" target="_self" title="QUOFIELD:def.29">Def29</a></i>;<br/>
<a NAME="E8:103"/><i><font color="Green" title="E83">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">h1</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="quofield.html#E5:103"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="quofield.html#E7:103"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>;<br/>
<a NAME="E9:103"/><i><font color="Green" title="E84">A7</font></i>: 
<font color="Maroon" title="c7">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">h1</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E5:103"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="quofield.html#E7:103"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="quofield.html#T57" target="_self" title="QUOFIELD:th.57">Th57</a></i>;<br/>
<a NAME="E10:103"/><i><font color="Green" title="E85">A8</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">h3</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c2">F</font><b> such that </b><br/><a NAME="E12:103"/><i><font color="Green" title="E86">A9</font></i>: 
( <font color="Maroon" title="c8">h3</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a> &amp; <font color="Maroon" title="c8">h3</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">f</font> &amp; ( for <font color="Olive" title="b1">h'</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c2">F</font>  st <font color="Olive" title="b1">h'</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a> &amp; <font color="Olive" title="b1">h'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">f</font> holds <br/><font color="Olive" title="b1">h'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c8">h3</font> ) )
 <b>by </b><i><a class="txt" href="quofield.html#E1:103"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="quofield.html#E2:103"><i><font color="Green" title="E79">A2</font></i></a>, <a class="ref" href="quofield.html#D29" target="_self" title="QUOFIELD:def.29">Def29</a></i>;<br/>
<font color="Maroon" title="c7">h2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">h1</font> = 
<font color="Maroon" title="c8">h3</font>
<b>by </b><i><a class="txt" href="quofield.html#E8:103"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="quofield.html#E9:103"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="quofield.html#E12:103"><i><font color="Green" title="E86">A9</font></i></a></i>
<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">F</font>
<b>by </b><i><a class="txt" href="quofield.html#E10:103"><i><font color="Green" title="E85">A8</font></i></a>, <a class="txt" href="quofield.html#E12:103"><i><font color="Green" title="E86">A9</font></i></a></i>
;<br/>
<a NAME="E14:103"/><b>then </b>
<font color="Maroon" title="c6">h1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#T37" title="FUNCT_2:th.37">FUNCT_2:37</a></i>;<br/>
<a NAME="E15:103"/><b>then </b><i><font color="Green" title="E87">A10</font></i>: 
<font color="Maroon" title="c6">h1</font> is <a href="quofield.html#V3" title="QUOFIELD:attr.3">RingMonomorphism</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E5:103"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="quofield.html#D23" target="_self" title="QUOFIELD:def.23">Def23</a></i>;<br/>
<a NAME="E16:103"/><i><font color="Green" title="E88">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">h2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f'</font>
 
<b>by </b><i><a class="txt" href="quofield.html#E5:103"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="quofield.html#E7:103"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>;<br/>
<a NAME="E17:103"/><i><font color="Green" title="E89">A12</font></i>: 
<font color="Maroon" title="c6">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">h2</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E5:103"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="quofield.html#E7:103"><i><font color="Green" title="E82">A5</font></i></a>, <a class="ref" href="quofield.html#T57" target="_self" title="QUOFIELD:th.57">Th57</a></i>;<br/>
<a NAME="E18:103"/><i><font color="Green" title="E90">A13</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c3">F'</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f'</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">h3</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">F'</font>,<font color="Maroon" title="c3">F'</font><b> such that </b><br/><a NAME="E20:103"/><i><font color="Green" title="E91">A14</font></i>: 
( <font color="Maroon" title="c9">h3</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a> &amp; <font color="Maroon" title="c9">h3</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f'</font> &amp; ( for <font color="Olive" title="b1">h'</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">F'</font>,<font color="Maroon" title="c3">F'</font>  st <font color="Olive" title="b1">h'</font> is <a href="quofield.html#V1" title="QUOFIELD:attr.1">RingHomomorphism</a> &amp; <font color="Olive" title="b1">h'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f'</font> holds <br/><font color="Olive" title="b1">h'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">h3</font> ) )
 <b>by </b><i><a class="txt" href="quofield.html#E1:103"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="quofield.html#E3:103"><i><font color="Green" title="E80">A3</font></i></a>, <a class="ref" href="quofield.html#D29" target="_self" title="QUOFIELD:def.29">Def29</a></i>;<br/>
<font color="Maroon" title="c6">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">h2</font> = 
<font color="Maroon" title="c9">h3</font>
<b>by </b><i><a class="txt" href="quofield.html#E16:103"><i><font color="Green" title="E88">A11</font></i></a>, <a class="txt" href="quofield.html#E17:103"><i><font color="Green" title="E89">A12</font></i></a>, <a class="txt" href="quofield.html#E20:103"><i><font color="Green" title="E91">A14</font></i></a></i>
<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">F'</font>
<b>by </b><i><a class="txt" href="quofield.html#E18:103"><i><font color="Green" title="E90">A13</font></i></a>, <a class="txt" href="quofield.html#E20:103"><i><font color="Green" title="E91">A14</font></i></a></i>
;<br/>
<a NAME="E22:103"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">h1</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="c3">F'</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T24" title="FUNCT_2:th.24">FUNCT_2:24</a></i>;<br/>
<a NAME="E23:103"/><b>then </b>
<font color="Maroon" title="c6">h1</font> is <a href="quofield.html#V2" title="QUOFIELD:attr.2">RingEpimorphism</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E5:103"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="quofield.html#D22" target="_self" title="QUOFIELD:def.22">Def22</a></i>;<br/>
<a NAME="E24:103"/><b>then </b>
<font color="Maroon" title="c6">h1</font> is <a href="quofield.html#V4" title="QUOFIELD:attr.4">RingIsomorphism</a>
 
<b>by </b><i><a class="txt" href="quofield.html#E15:103"><i><font color="Green" title="E87">A10</font></i></a>, <a class="ref" href="quofield.html#D24" target="_self" title="QUOFIELD:def.24">Def24</a></i>;<br/>
<b>hence </b><a NAME="E25:103"/>
<font color="Maroon" title="c2">F</font> <a href="quofield.html#R2" title="QUOFIELD:pred.2">is_ringisomorph_to</a> <font color="Maroon" title="c3">F'</font>
 <b>by </b><i><a class="ref" href="quofield.html#D26" target="_self" title="QUOFIELD:def.26">Def26</a></i>;<br/>


</div>
