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;

theorem Th24:
  a <> 0c implies Carrier (a * L) = Carrier L
proof
  set T = {u : (a * L).u <> 0c};
  set S = {v : L.v <> 0c};
  assume
A1: a <> 0c;
  T = S
  proof
    thus T c= S
    proof
      let x be object;
      assume x in T;
      then consider u such that
A2:   x = u and
A3:   (a * L).u <> 0c;
      (a * L).u = a * L.u by Def9;
      then L.u <> 0c by A3;
      hence thesis by A2;
    end;
    let x be object;
    assume x in S;
    then consider v such that
A4: x = v & L.v <> 0c;
    (a * L).v = a * L.v by Def9;
    hence thesis by A1,A4;
  end;
  hence thesis;
end;
