reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem
  for V,W be finite-dimensional RealLinearSpace
    st dim V <> 0
  holds
    dim V = dim W
      iff
    ex T be LinearOperator of V, W st T is bijective
  proof
    let V,W be finite-dimensional RealLinearSpace;
    assume A1: dim V <> 0;
    hereby
      assume
      A2: dim V = dim W;

      consider T1 be LinearOperator of V,
      RealVectSpace Seg dim V such that
      A3: T1 is bijective by A1,Th89;

      consider T2 be LinearOperator of W,
      RealVectSpace Seg dim V such that
      A4: T2 is bijective by A1,A2,Th89;

      consider S be LinearOperator of RealVectSpace(Seg(dim V)),W such that
      A5: S = T2 " & S is one-to-one onto by A4,Th85;
      set T = S * T1;
      reconsider T as LinearOperator of V,W by Th86;
      T is bijective by A3,A5,FINSEQ_4:85;
      hence ex T be LinearOperator of V,W
      st T is bijective;
    end;
    assume ex T be LinearOperator of V, W st T is bijective;
    hence dim V = dim W by Th88;
  end;
