let F1, F2 be Function of (TopSpaceMetr M),R^1; :: thesis: ( ( for x being Point of M holds F1 . x = lower_bound ((dist x) .: P) ) & ( for x being Point of M holds F2 . x = lower_bound ((dist x) .: P) ) implies F1 = F2 )
assume A12: for y being Point of M holds F1 . y = lower_bound ((dist y) .: P) ; :: thesis: ( ex x being Point of M st not F2 . x = lower_bound ((dist x) .: P) or F1 = F2 )
assume A13: for y being Point of M holds F2 . y = lower_bound ((dist y) .: P) ; :: thesis: F1 = F2
for y being set st y in the carrier of (TopSpaceMetr M) holds
F1 . y = F2 . y

proof

let y be set ; :: thesis: ( y in the carrier of (TopSpaceMetr M) implies F1 . y = F2 . y )
A14: TopSpaceMetr M = TopStruct(# the carrier of M,(Family_open_set M) #) by PCOMPS_1:def 5;
assume y in the carrier of (TopSpaceMetr M) ; :: thesis: F1 . y = F2 . y
then reconsider y = y as Point of M by A14;
F1 . y = lower_bound ((dist y) .: P) by A12
.= F2 . y by A13 ;
hence F1 . y = F2 . y ; :: thesis: verum

end;

hence F1 = F2 by FUNCT_2:12; :: thesis: verum