reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;

theorem Th42:
  a <> 0 implies Carrier(a * L) = Carrier(L)
proof
  set T = {u : (a * L).u <> 0};
  set S = {v : L.v <> 0};
  assume
A1: a <> 0;
  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 <> 0;
      (a * L).u = a * L.u by Def11;
      then L.u <> 0 by A3;
      hence thesis by A2;
    end;
    let x be object;
    assume x in S;
    then consider v such that
A4: x = v and
A5: L.v <> 0;
    (a * L).v = a * L.v by Def11;
    then (a * L).v <> 0 by A1,A5,XCMPLX_1:6;
    hence thesis by A4;
  end;
  hence thesis;
end;
