let s1, s2 be Real_Sequence; :: thesis: ( ( for n being Nat holds s1 . n = a #Q (s . n) ) & ( for n being Nat holds s2 . n = a #Q (s . n) ) implies s1 = s2 )

assume that

A2: for n being Nat holds s1 . n = a #Q (s . n) and

A3: for n being Nat holds s2 . n = a #Q (s . n) ; :: thesis: s1 = s2

assume that

A2: for n being Nat holds s1 . n = a #Q (s . n) and

A3: for n being Nat holds s2 . n = a #Q (s . n) ; :: thesis: s1 = s2

now :: thesis: for n being Element of NAT holds s1 . n = s2 . n

hence
s1 = s2
by FUNCT_2:63; :: thesis: verumlet n be Element of NAT ; :: thesis: s1 . n = s2 . n

thus s1 . n = a #Q (s . n) by A2

.= s2 . n by A3 ; :: thesis: verum

end;thus s1 . n = a #Q (s . n) by A2

.= s2 . n by A3 ; :: thesis: verum