for E, L being Z_Module
for I being Subset of L
for J being Subset of E st I = J & ModuleStr(# the carrier of L, the addF of L, the ZeroF of L, the lmult of L #) = ModuleStr(# the carrier of E, the addF of E, the ZeroF of E, the lmult of E #) & I is linearly-independent holds
J is linearly-independent
proof
let E, L be Z_Module; :: thesis: for I being Subset of L
for J being Subset of E st I = J & ModuleStr(# the carrier of L, the addF of L, the ZeroF of L, the lmult of L #) = ModuleStr(# the carrier of E, the addF of E, the ZeroF of E, the lmult of E #) & I is linearly-independent holds
J is linearly-independent

let I be Subset of L; :: thesis: for J being Subset of E st I = J & ModuleStr(# the carrier of L, the addF of L, the ZeroF of L, the lmult of L #) = ModuleStr(# the carrier of E, the addF of E, the ZeroF of E, the lmult of E #) & I is linearly-independent holds
J is linearly-independent

let J be Subset of E; :: thesis: ( I = J & ModuleStr(# the carrier of L, the addF of L, the ZeroF of L, the lmult of L #) = ModuleStr(# the carrier of E, the addF of E, the ZeroF of E, the lmult of E #) & I is linearly-independent implies J is linearly-independent )
assume that
A1: I = J and
A3: ModuleStr(# the carrier of L, the addF of L, the ZeroF of L, the lmult of L #) = ModuleStr(# the carrier of E, the addF of E, the ZeroF of E, the lmult of E #) and
A2: I is linearly-independent ; :: thesis:
for K being Linear_Combination of J st Sum K = 0. E holds
Carrier K = {}
proof
let K be Linear_Combination of J; :: thesis: ( Sum K = 0. E implies Carrier K = {} )
assume P0: Sum K = 0. E ; :: thesis:
reconsider H = K as Linear_Combination of I by ;
P1: Carrier K = Carrier H by A3;
Sum H = 0. L by ;
hence Carrier K = {} by ; :: thesis: verum
end;
hence J is linearly-independent by VECTSP_7:def 1; :: thesis: verum
end;
hence for L, E being Z_Module
for I being Subset of L
for J being Subset of E st ModuleStr(# the carrier of L, the addF of L, the ZeroF of L, the lmult of L #) = ModuleStr(# the carrier of E, the addF of E, the ZeroF of E, the lmult of E #) & I = J holds
( I is linearly-independent iff J is linearly-independent ) ; :: thesis: verum