let s, t be Real_Sequence; :: thesis: ( ( for n being Element of NAT holds s . n = dist (seq . n),x ) & ( for n being Element of NAT holds t . n = dist (seq . n),x ) implies s = t )
assume that
A2: for n being Element of NAT holds s . n = dist (seq . n),x and
A3: for n being Element of NAT holds t . n = dist (seq . n),x ; :: thesis: s = t

now

let n be Element of NAT ; :: thesis: s . n = t . n
s . n = dist (seq . n),x by A2;
hence s . n = t . n by A3; :: thesis: verum

end;

hence s = t by FUNCT_2:113; :: thesis: verum