
theorem
  for X,Y be RealLinearSpace,
      L be LinearOperator of X,Y
  st L is isomorphism holds
  ex K be LinearOperator of Y,X st K = L" & K is isomorphism
  proof
    let X,Y be RealLinearSpace,
        L be LinearOperator of X,Y;
    assume
    A1: L is isomorphism; then
    A2: L is one-to-one & L is onto;
    reconsider K = L" as Function of Y,X by A2,FUNCT_2:25;
    A3: dom L = the carrier of X by FUNCT_2:def 1;
    A4: K is additive
    proof
      let x,y be Point of Y;
      consider a be Point of X such that
      A5: x = L.a by A2,FUNCT_2:113;
      consider b be Point 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;
    K is homogeneous
    proof
      let x be Point of Y, r be Real;
      consider a be Point of X such that
      A10: x = L.a by A2,FUNCT_2:113;
      A11: K.x = a by A1,A3,A10,FUNCT_1:34;
      A12: r*x = L.(r*a) by A10,LOPBAN_1:def 5;
      thus K.(r*x) = r * K.x by A1,A3,A11,A12,FUNCT_1:34;
    end; then
    reconsider K as LinearOperator of Y, X by A4;
    take K;
    K is onto by A1,A3,FUNCT_1:33;
    hence thesis by A1,FUNCT_1:40;
  end;
