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In the next Subdivision I consider a logical transformation ''F'' that has the concrete type ''F'' : [''u'', ''v''] → [''x'', ''y''] and the abstract type ''F'' : ['''B'''²] → ['''B'''²].  From the standpoint of propositional calculus, the task of understanding such a transformation is naturally approached by parsing it into component maps with 1-dimensional ranges, as follows:

In the next Subdivision I consider a logical transformation ''F'' that has the concrete type ''F'' : [''u'', ''v''] → [''x'', ''y''] and the abstract type ''F'' : ['''B'''²] → ['''B'''²].  From the standpoint of propositional calculus, the task of understanding such a transformation is naturally approached by parsing it into component maps with 1-dimensional ranges, as follows:

<pre>

<br><font face="courier new">

o-----------------------------------------------------------o

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"

|                                                           |

|   F  =   <f, g> =  <F_1, F_2> : [u, v] -> [x, y]   |

{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

|                                                           |

| align="left" | ''F''

|   where      f   =     F_1    : [u, v] -> [x]       |

| =

|                                                           |

| ‹''f'', ''g''›

|   and        g   =     F_2    : [u, v] -> [y]       |

|                                                           |

| ‹''F''₁, ''F''₂›

| :

</pre>

| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>

| <nowiki>[</nowiki>''x'', ''y''<nowiki>]</nowiki>

|-

| align="left" colspan="2" | where

| ''f''

| =

| :

| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>

| <nowiki>[</nowiki>''x''<nowiki>]</nowiki>

|-

| align="left" colspan="2" | and

| ''g''

| =

| :

| <nowiki>[</nowiki>''u'', ''v''<nowiki>]</nowiki>

| <nowiki>[</nowiki>''y''<nowiki>]</nowiki>

|}

|}

</font><br>

Then one tackles the separate components, now viewed as propositions ''F''_''i''&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''', one at a time.  At the completion of this analytic phase, one returns to the task of synthesizing all of these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation.  (Very often, of course, in tangling with refractory cases, one never gets as far as the beginning again.)

Then one tackles the separate components, now viewed as propositions ''F''_''i''&nbsp;:&nbsp;''U''&nbsp;&rarr;&nbsp;'''B''', one at a time.  At the completion of this analytic phase, one returns to the task of synthesizing all of these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation.  (Very often, of course, in tangling with refractory cases, one never gets as far as the beginning again.)