let f1, f2 be Function; :: thesis: ( ( for a being Element of L holds

( dom f1 = the carrier of L & f1 . a = { F where F is Filter of L : ( F in F_primeSet L & a in F ) } ) ) & ( for a being Element of L holds

( dom f2 = the carrier of L & f2 . a = { F where F is Filter of L : ( F in F_primeSet L & a in F ) } ) ) implies f1 = f2 )

assume that

A2: for a being Element of L holds

( dom f1 = the carrier of L & f1 . a = { F where F is Filter of L : ( F in F_primeSet L & a in F ) } ) and

A3: for a being Element of L holds

( dom f2 = the carrier of L & f2 . a = { F where F is Filter of L : ( F in F_primeSet L & a in F ) } ) ; :: thesis: f1 = f2

( dom f1 = the carrier of L & f1 . a = { F where F is Filter of L : ( F in F_primeSet L & a in F ) } ) ) & ( for a being Element of L holds

( dom f2 = the carrier of L & f2 . a = { F where F is Filter of L : ( F in F_primeSet L & a in F ) } ) ) implies f1 = f2 )

assume that

A2: for a being Element of L holds

( dom f1 = the carrier of L & f1 . a = { F where F is Filter of L : ( F in F_primeSet L & a in F ) } ) and

A3: for a being Element of L holds

( dom f2 = the carrier of L & f2 . a = { F where F is Filter of L : ( F in F_primeSet L & a in F ) } ) ; :: thesis: f1 = f2

now :: thesis: for x being object st x in the carrier of L holds

f1 . x = f2 . x

hence
f1 = f2
by A2, A3; :: thesis: verumf1 . 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 in F_primeSet L & a in F ) } by A2

.= f2 . x by A3 ; :: thesis: verum

end;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 in F_primeSet L & a in F ) } by A2

.= f2 . x by A3 ; :: thesis: verum