reserve x,y for set,
  i for Nat;
reserve V for non empty CLSStruct,
  u,v,v1,v2,v3 for VECTOR of V,
  A for Subset of V,
  l, l1, l2 for C_Linear_Combination of A,
  x,y,y1,y2 for set,
  a,b for Complex,
  F for FinSequence of the carrier of V,
  f for Function of the carrier of V, COMPLEX;
reserve K,L,L1,L2,L3 for C_Linear_Combination of V;
reserve e,e1,e2 for Element of C_LinComb V;

theorem Th60:
  for V being vector-distributive scalar-distributive scalar-associative
  scalar-unital non empty CLSStruct, M
  being Subset of V, z1,z2 being Complex holds (z1 + z2)*M c= z1*M + z2*M
proof
  let V be vector-distributive scalar-distributive scalar-associative
  scalar-unital non empty CLSStruct;
  let M be Subset of V;
  let z1,z2 be Complex;
  for x being VECTOR of V st x in (z1+z2)*M holds x in z1*M + z2*M
  proof
    let x be VECTOR of V;
    assume x in (z1+z2)*M;
    then consider w be VECTOR of V such that
A1: x = (z1 + z2)*w and
A2: w in M;
A3: z2*w in z2*M by A2;
    x = z1*w + z2*w & z1*w in z1*M by A1,A2,CLVECT_1:def 3;
    hence thesis by A3;
  end;
  hence thesis;
end;
