 reserve X,Y,Z,E,F,G,S,T for RealLinearSpace;

theorem LM020:
  for I be LinearOperator of S, T st I is bijective holds
   ex J be LinearOperator of T, S st J = I" & J is bijective
  proof
    let I be LinearOperator of S, T;
    assume
    A1: I is bijective; then
a1: I is one-to-one onto; then
    A2: rng I = the carrier of T & dom I = the carrier of S by FUNCT_2:def 1;
    A3: rng I = dom(I") & dom I = rng(I") by A1,FUNCT_1:33; then
    reconsider J = I" as Function of T,S by A2,FUNCT_2:1;
    A4: for v, w be Point of T holds J.(v+w) = J.v + J.w
    proof
      let v, w be Point of T;
      consider t be Point of S such that
      A5: v = I.t by FUNCT_2:113,a1;
      consider s be Point of S such that
      A6: w = I.s by FUNCT_2:113,a1;
      A7: J.(v+w) = J.(I.(t+s)) by A5,A6,VECTSP_1:def 20
      .= t+s by A1,A2,FUNCT_1:34;
      J.w = s by A1,A2,A6,FUNCT_1:34;
      hence thesis by A1,A2,A5,A7,FUNCT_1:34;
    end;
    for v be Point of T, r be Real holds J.(r*v) = r*(J.v)
    proof
      let v be Point of T,r be Real;
      consider t be Point of S such that
      A9: v = I.t by FUNCT_2:113,a1;
      J.(r*v) = J.(I.(r*t)) by A9,LOPBAN_1:def 5
      .= r*t by A1,A2,FUNCT_1:34;
      hence thesis by A1,A2,A9,FUNCT_1:34;
    end;
    then reconsider J as LinearOperator of T,S
      by A4,LOPBAN_1:def 5,VECTSP_1:def 20;
    take J;
    thus J = I";
    J is one-to-one onto by A1,A3,FUNCT_2:def 1;
    hence J is bijective;
  end;
