reserve x,y for object, X,Y,Z for set;
reserve a,b for Real;
reserve k for Element of NAT;
reserve V for RealLinearSpace;
reserve W1,W2,W3 for Subspace of V;
reserve v,v1,v2,u for VECTOR of V;
reserve A,B,C for Subset of V;
reserve T for finite Subset of V;
reserve L,L1,L2 for Linear_Combination of V;
reserve l for Linear_Combination of A;
reserve F,G,H for FinSequence of the carrier of V;
reserve f,g for Function of the carrier of V, REAL;
reserve p,q,r for FinSequence;
reserve M for non empty set;
reserve CF for Choice_Function of M;
reserve l0 for Linear_Combination of {}(the carrier of V);

theorem Th20:
  A c= B implies Lin(A) is Subspace of Lin(B)
proof
  assume
A1: A c= B;
  now
    let v;
    assume v in Lin(A);
    then consider l such that
A2: v = Sum(l) by Th14;
    reconsider l as Linear_Combination of B by A1,RLVECT_2:21;
    Sum(l) = v by A2;
    hence v in Lin(B) by Th14;
  end;
  hence thesis by RLSUB_1:29;
end;
