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These two diagrams show the situation at an angle of attack of 19 degrees, which is right after stall.
In the aerofoil without flaps you can see that flow separation has commenced almost to the leading edge.
On both sides, however, an area of relatively high lift generation prevails due to the effect of the trailing vortices.
If you look at the lower diagram with the lift distribution for the aerofoil with the flaps attached, it is obvious that the eddy, and therefore flow separation, is being prevented from spreading further towards the leading edge by the flaps.
The eddy is restricted to the hind part of the aerofoil, where not to much lift is generated anyway.
As a result, in front of the flaps the low pressure and therefore lift generation can remain at a much higher level than it were without flaps.