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: J is linearly-independent
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: Carrier K = {}
reconsider H = K as Linear_Combination of I by A1, A3, LmEMDetX541;
P1: Carrier K = Carrier H by A3;
Sum H = 0. L by A3, P0, LmEMDetX543;
hence Carrier K = {} by A2, P1, VECTSP_7:def 1; :: 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