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
now :: thesis: for x being object st x in the carrier of BL holds
f1 . 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;
hence f1 = f2 by ; :: thesis: verum