let s1, s2 be ManySortedSet of NAT ; :: thesis: ( ( for n being Nat holds s1 . n = s . (n + k) ) & ( for n being Nat holds s2 . n = s . (n + k) ) implies s1 = s2 )
assume that
A3: for n being Nat holds s1 . n = s . (n + k) and
A4: for n being Nat holds s2 . n = s . (n + k) ; :: thesis: s1 = s2

now :: thesis: for n being object st n in NAT holds
s1 . n = s2 . n

let n be object ; :: thesis: ( n in NAT implies s1 . n = s2 . n )
assume n in NAT ; :: thesis: s1 . n = s2 . n
then reconsider nn = n as Element of NAT ;
thus s1 . n = s . (nn + k) by A3
.= s2 . n by A4 ; :: thesis: verum

end;

hence s1 = s2 by PBOOLE:3; :: thesis: verum