let g, h be PartFunc of X,COMPLEX; :: thesis: ( dom g = dom (f . 0) & ( for x being Element of X st x in dom g holds
g . x = lim (f # x) ) & dom h = dom (f . 0) & ( for x being Element of X st x in dom h holds
h . x = lim (f # x) ) implies g = h )
assume that
A4: dom g = dom (f . 0) and
A5: for x being Element of X st x in dom g holds
g . x = lim (f # x) ; :: thesis: ( not dom h = dom (f . 0) or ex x being Element of X st
( x in dom h & not h . x = lim (f # x) ) or g = h )
assume that
A6: dom h = dom (f . 0) and
A7: for x being Element of X st x in dom h holds
h . x = lim (f # x) ; :: thesis: g = h

now :: thesis: for x being Element of X st x in dom g holds
g . x = h . x

let x be Element of X; :: thesis: ( x in dom g implies g . x = h . x )
assume A8: x in dom g ; :: thesis: g . x = h . x
then g . x = lim (f # x) by A5;
hence g . x = h . x by A4, A6, A7, A8; :: thesis: verum

end;

hence g = h by A4, A6, PARTFUN1:5; :: thesis: verum