The Rules Compendium does not prohibit spellcasting within an area of antimagic but does say a spell's effect is either suppressed or negated if a spell's point of origin is within an area of antimagic...
The Rules Compendium on Antimagic says
Spells don’t function in an antimagic area, but an antimagic area
doesn’t block line of effect. If a spell’s point of origin is inside an antimagic area, that spell is entirely suppressed. When a spell’s point of origin is located outside an antimagic area, but part of that spell’s area overlaps the antimagic area, that spell’s effect is suppressed where the two areas overlap. Time elapsed within an antimagic area still counts against a spell’s duration.
If an instantaneous spell is entirely suppressed, that spell is effectively canceled. (It’s suppressed, and its duration instantaneously expires.) An instantaneous area spell is only entirely suppressed and effectively canceled if its point of origin is within the antimagic area. Otherwise it works like any other area spell that has a point of origin outside the antimagic area—only where its area overlaps the antimagic area is its effect is suppressed
(and effectively canceled). (11)
So the caster of the 6th-level Sor/Wiz spell antimagic field [abjur] (PH 200) (in addition to anyone who joins the caster within the field) has all magic suppressed while the field remains, but the caster doesn't actually lose the ability to cast spells or anything. Instead, spells that are cast are suppressed while the field remains or, if instantaneous, "effectively cancelled" if their points of origin are within the field.1
Thus, as per the illustration, when a caster enters one of those antimagic fields and casts the 9th-level Sor/Wiz spell Mordenkainen's disjunction [abjur] (PH 255), that caster picks the disjunction spell's point of origin. If he picks as the spell's point of origin a grid intersection covered by an antimagic field, the disjunction spell will fail. If the caster picks a grid intersection in the midst of the prismatic sphere, the spell will also fail as the caster lacks both line of sight and line of effect beyond the sphere.
The Range: Close of the spell Mordenkainen's disjunction means the caster's far better off picking as the point of origin for the disjunction a grid intersection above this excessively complicated, high-powered, volatile concentration of overlapping spell areas and seeing if the disjunction's 40-ft.-radius burst takes out any of the antimagic fields (1% per caster level) in addition to dispelling the prismatic sphere.
...But the Dungeon Master's Guide strongly implies that spells can't be cast spells in an area of antimagic
The Dungeon Master's Guide on Antimagic says
No supernatural ability, spell-like ability, or spell works in an area of antimagic (but extraordinary abilities still work). (290)
Lawyering the word works here becomes the issue,2 but later authors and editors likely believed this line (and, presumably, other evidence or house rules or whatever) significant enough to engender the creation of...
- the feat Initiate of Mystra (Player's Guide to Faerûn 81): "You can... cast spells even within... an antimagic field [by making] a successful caster level check against a DC equal to 11 + the caster level of the antimagic field."
- the 9th-level Sor/Wiz spell invoke magic [evoc] (Lords of Madness 212): "Within the area of an antimagic field, a dead magic area, or a null-magic plane... you [can] cast a single spell of 4th level or lower, which then ends [this] spell’s effect."
While both these artifacts are 3.5, both Player's Guide and Lords predate the Rules Compendium. Further, keep in mind that the entry on Antimagic from the Dungeon Master's Guide (2012) is verbatim the Antimagic entry from the Dungeon Master's Guide (2000) for Dungeons and Dragons, Third Edition, which had folks reading antimagic field as blocking line of effect.
Avoid problems by not using areas of antimagic
You can read the fights that the spell antimagic field creates in these threads from 2006,3 2009, 2010, 2011 (Warning: 9 pages!), 2011 (Warning: for Pathfinder—and the same problems), 2012, and 2013. These are merely a handful of the many threads just about casting spells in an antimagic field; there exist many other threads about the vast number of other antimagic field issues.
Thus, while picking either of the above readings (or developing another) is a solution if a campaign must incorporate areas of antimagic, I suggest not using areas of antimagic at all. Almost any discussion will lead to an argument and, usually, less fun for someone. It's an effect that makes no one happy, and almost any campaign is better off without it.4
1 The Rules Compendium on Out of Sight? No Problem! says, "For ranged attacks and other effects that originate from a creature, the point of origin is any corner of any square that the creature occupies" (80).
2 The Rules Compendium also says, "No supernatural ability or spell-like ability works in an antimagic area" (as does DMG 90). Then, immediately after, the RC adds, "Their effects are affected like spells" (11). This is singularly unhelpful.
3 Noteworthy in predating the Rules Compendium and citing the Third Edition FAQ that apparently said antimagic field does (or, that is, did), in fact, block line of effect.
4 Beholders can do other things with their central eyes.