let f, g be Function of the carrier of s,NAT ; :: thesis: ( ( for x being type of s holds f . x = card (dom (repr_of x)) ) & ( for x being type of s holds g . x = card (dom (repr_of x)) ) implies f = g )
assume that
A1: for x being type of s holds f . x = card (dom (repr_of x)) and
A2: for x being type of s holds g . x = card (dom (repr_of x)) ; :: thesis: f = g

now

let c be Element of s; :: thesis: f . c = g . c
thus f . c = card (dom (repr_of c)) by A1
.= g . c by A2 ; :: thesis: verum

end;

hence f = g by FUNCT_2:113; :: thesis: verum