let f1, f2 be Function; :: thesis: ( dom f1 = PARTITIONS Y & ( for x being object st x in PARTITIONS Y holds
ex PA being a_partition of Y st
( PA = x & f1 . x = ERl PA ) ) & dom f2 = PARTITIONS Y & ( for x being object st x in PARTITIONS Y holds
ex PA being a_partition of Y st
( PA = x & f2 . x = ERl PA ) ) implies f1 = f2 )
assume that
A2: dom f1 = PARTITIONS Y and
A3: for x being object st x in PARTITIONS Y holds
ex PA being a_partition of Y st
( PA = x & f1 . x = ERl PA ) and
A4: dom f2 = PARTITIONS Y and
A5: for x being object st x in PARTITIONS Y holds
ex PA being a_partition of Y st
( PA = x & f2 . x = ERl PA ) ; :: thesis: f1 = f2
for z being object st z in PARTITIONS Y holds
f1 . z = f2 . z

proof

let x be object ; :: thesis: ( x in PARTITIONS Y implies f1 . x = f2 . x )
assume x in PARTITIONS Y ; :: thesis: f1 . x = f2 . x
then ( ex PA being a_partition of Y st
( PA = x & f1 . x = ERl PA ) & ex PA being a_partition of Y st
( PA = x & f2 . x = ERl PA ) ) by A3, A5;
hence f1 . x = f2 . x ; :: thesis: verum

end;

hence f1 = f2 by A2, A4, FUNCT_1:2; :: thesis: verum