let f1, f2 be Function; :: thesis: ( dom f1 = the carrier of BL & ( for a being Element of BL holds f1 . a = { UF where UF is Filter of BL : ( UF is being_ultrafilter & a in UF ) } ) & dom f2 = the carrier of BL & ( for a being Element of BL holds f2 . a = { UF where UF is Filter of BL : ( UF is being_ultrafilter & a in UF ) } ) implies f1 = f2 )

assume that

A3: ( dom f1 = the carrier of BL & ( for a being Element of BL holds f1 . a = { F where F is Filter of BL : ( F is being_ultrafilter & a in F ) } ) ) and

A4: ( dom f2 = the carrier of BL & ( for a being Element of BL holds f2 . a = { F where F is Filter of BL : ( F is being_ultrafilter & a in F ) } ) ) ; :: thesis: f1 = f2

assume that

A3: ( dom f1 = the carrier of BL & ( for a being Element of BL holds f1 . a = { F where F is Filter of BL : ( F is being_ultrafilter & a in F ) } ) ) and

A4: ( dom f2 = the carrier of BL & ( for a being Element of BL holds f2 . a = { F where F is Filter of BL : ( F is being_ultrafilter & a in F ) } ) ) ; :: thesis: f1 = f2

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

f1 . x = f2 . x

hence
f1 = f2
by A3, A4, FUNCT_1:2; :: thesis: verumf1 . x = f2 . x

let x be object ; :: thesis: ( x in the carrier of BL implies f1 . x = f2 . x )

assume x in the carrier of BL ; :: thesis: f1 . x = f2 . x

then reconsider a = x as Element of BL ;

thus f1 . x = { F where F is Filter of BL : ( F is being_ultrafilter & a in F ) } by A3

.= f2 . x by A4 ; :: thesis: verum

end;assume x in the carrier of BL ; :: thesis: f1 . x = f2 . x

then reconsider a = x as Element of BL ;

thus f1 . x = { F where F is Filter of BL : ( F is being_ultrafilter & a in F ) } by A3

.= f2 . x by A4 ; :: thesis: verum