let s1, s2 be Real_Sequence; :: thesis: ( ( for m being Element of NAT holds s1 . m = a |^ m ) & ( for m being Element of NAT holds s2 . m = a |^ m ) implies s1 = s2 )
assume that
A1: for n being Element of NAT holds s1 . n = a |^ n and
A2: for n being Element of NAT holds s2 . n = a |^ n ; :: thesis: s1 = s2
for n being Element of NAT holds s1 . n = s2 . n

proof

let n be Element of NAT ; :: thesis: s1 . n = s2 . n
thus s1 . n = a |^ n by A1
.= s2 . n by A2 ; :: thesis: verum

end;

hence s1 = s2 by FUNCT_2:113; :: thesis: verum