let n be Nat; :: thesis:
let Ar be Subset of (REAL-NS n); :: thesis:
let At be Subset of (TOP-REAL n); :: thesis:
assume A1: Ar = At ; :: thesis:

hereby :: thesis:

assume A2: Ar is linearly-independent ; :: thesis:

let L be Linear_Combination of At; :: thesis:
reconsider L1 = L as Linear_Combination of REAL-NS n by Th11;
A3: Carrier L1 = Carrier L ;
assume Sum L = 0. (TOP-REAL n) ; :: thesis:
then A4: 0. (REAL-NS n) = Sum L by Th6
.= Sum L1 by Th23 ;
L1 is Linear_Combination of Ar by A1, A3, RLVECT_2:def 6;
hence Carrier L = {} by A2, A4, RLVECT_3:def 1; :: thesis:

end;

hence At is linearly-independent by RLVECT_3:def 1; :: thesis:

end;

assume A5: At is linearly-independent ; :: thesis:

let L be Linear_Combination of Ar; :: thesis:
reconsider L1 = L as Linear_Combination of TOP-REAL n by Th11;
A6: Carrier L1 = Carrier L ;
reconsider L1 = L as Linear_Combination of At by A1, A6, RLVECT_2:def 6;
assume Sum L = 0. (REAL-NS n) ; :: thesis:
then 0. (TOP-REAL n) = Sum L by Th6
.= Sum L1 by Th23 ;
hence Carrier L = {} by A5, RLVECT_3:def 1; :: thesis:

end;

hence Ar is linearly-independent by RLVECT_3:def 1; :: thesis: