let f1, f2 be Function; :: thesis: ( dom f1 = the carrier of L & ( for a being Element of L holds f1 . a = { F where F is Filter of L : ( F is prime & a in F ) } ) & dom f2 = the carrier of L & ( for a being Element of L holds f2 . a = { F where F is Filter of L : ( F is prime & a in F ) } ) implies f1 = f2 )
assume that
A3: ( dom f1 = the carrier of L & ( for a being Element of L holds f1 . a = { F where F is Filter of L : ( F is prime & a in F ) } ) ) and
A4: ( dom f2 = the carrier of L & ( for a being Element of L holds f2 . a = { F where F is Filter of L : ( F is prime & a in F ) } ) ) ; :: thesis: f1 = f2

now :: thesis: for x being object st x in the carrier of L holds
f1 . x = f2 . x

let x be object ; :: thesis: ( x in the carrier of L implies f1 . x = f2 . x )
assume x in the carrier of L ; :: thesis: f1 . x = f2 . x
then reconsider a = x as Element of L ;
thus f1 . x = { F where F is Filter of L : ( F is prime & a in F ) } by A3
.= f2 . x by A4 ; :: thesis: verum

end;

hence f1 = f2 by A3, A4, FUNCT_1:2; :: thesis: verum