let p be prime Element of INT.Ring; :: thesis: for V being free Z_Module
for I being Basis of V
for IQ being Subset of (Z_MQ_VectSp (V,p)) st IQ = { (ZMtoMQV (V,p,u)) where u is Vector of V : u in I } holds
IQ is Basis of (Z_MQ_VectSp (V,p))

let V be free Z_Module; :: thesis: for I being Basis of V
for IQ being Subset of (Z_MQ_VectSp (V,p)) st IQ = { (ZMtoMQV (V,p,u)) where u is Vector of V : u in I } holds
IQ is Basis of (Z_MQ_VectSp (V,p))

let I be Basis of V; :: thesis: for IQ being Subset of (Z_MQ_VectSp (V,p)) st IQ = { (ZMtoMQV (V,p,u)) where u is Vector of V : u in I } holds
IQ is Basis of (Z_MQ_VectSp (V,p))

let IQ be Subset of (Z_MQ_VectSp (V,p)); :: thesis: ( IQ = { (ZMtoMQV (V,p,u)) where u is Vector of V : u in I } implies IQ is Basis of (Z_MQ_VectSp (V,p)) )
assume A1: IQ = { (ZMtoMQV (V,p,u)) where u is Vector of V : u in I } ; :: thesis: IQ is Basis of (Z_MQ_VectSp (V,p))
A2: IQ is linearly-independent by ;
for vq being Element of (Z_MQ_VectSp (V,p)) holds vq in Lin IQ
proof
let vq be Element of (Z_MQ_VectSp (V,p)); :: thesis: vq in Lin IQ
consider v being Vector of V such that
A3: vq = ZMtoMQV (V,p,v) by Th22;
I is base ;
then Lin I = ModuleStr(# the carrier of V, the addF of V, the ZeroF of V, the lmult of V #) ;
then consider l being Linear_Combination of I such that
A4: v = Sum l by ;
thus vq in Lin IQ by A1, A4, A3, Th34; :: thesis: verum
end;
then Lin IQ = ModuleStr(# the carrier of (Z_MQ_VectSp (V,p)), the addF of (Z_MQ_VectSp (V,p)), the ZeroF of (Z_MQ_VectSp (V,p)), the lmult of (Z_MQ_VectSp (V,p)) #) by VECTSP_4:32;
then IQ is base by A2;
hence IQ is Basis of (Z_MQ_VectSp (V,p)) ; :: thesis: verum