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 Th61:
  for V being non empty CLSStruct, M,N being Subset of V, z being
  Complex st M c= N holds z*M c= z*N
proof
  let V be non empty CLSStruct;
  let M,N be Subset of V;
  let z be Complex;
  assume
A1: M c= N;
  now
    let x be VECTOR of V;
    assume x in z*M;
    then ex w be VECTOR of V st x = z*w & w in M;
    hence x in z*N by A1;
  end;
  hence thesis;
end;
