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 Th85:
  for X, Y be RealLinearSpace,
         L be LinearOperator of X,Y
    st L is bijective
  holds
    ex K be LinearOperator of Y,X
    st K = L" & K is one-to-one onto
  proof
    let X, Y be RealLinearSpace;
    let L be LinearOperator of X,Y;
    assume
    A1: L is bijective;
    then
    A2: rng L = the carrier of Y by FUNCT_2:def 3;
    then reconsider K = L" as Function of the carrier of Y,the carrier of X
      by A1,FUNCT_2:25;
    A3: dom L = the carrier of X by FUNCT_2:def 1;
    A4: K is additive
    proof
      let x, y be Element of Y;
      consider a be Element of X such that
      A5: x = L . a by A2,FUNCT_2:113;
      consider b be Element of X such that
      A6: y = L . b by A2,FUNCT_2:113;
      A7: K . x = a by A1,A3,A5,FUNCT_1:34;
      A8: K . y = b by A1,A3,A6,FUNCT_1:34;
      x + y = L . (a + b) by A5,A6,VECTSP_1:def 20;
      hence K . (x + y) = (K . x) + (K . y) by A1,A3,A7,A8,FUNCT_1:34;
    end;

    for x be VECTOR of Y
    for r be Real
    holds K . (r * x) = r * (K . x)
    proof
      let x be VECTOR of Y;
      let r be Real;

      consider a be VECTOR of X such that
      A9: x = L . a by A2,FUNCT_2:113;
      A10: K . x = a by A1,A3,A9,FUNCT_1:34;
      r * x = L . (r * a) by A9,LOPBAN_1:def 5;
      hence K . (r * x) = r * (K . x) by A1,A3,A10,FUNCT_1:34;
    end;
    then reconsider K as LinearOperator of Y,X by A4,LOPBAN_1:def 5;
    take K;
    rng K = the carrier of X by A1,A3,FUNCT_1:33;
    hence K = L " & K is one-to-one onto by A1,FUNCT_2:def 3;
  end;
