
theorem Th1:
  for X,Y be RealLinearSpace,
      T be LinearOperator of X,Y
    st T is bijective
    holds T" is LinearOperator of Y,X & rng(T") = the carrier of X
  proof
    let X,Y be RealLinearSpace,
        T be LinearOperator of X,Y;
    assume A1: T is bijective;
A2:rng T = the carrier of Y by A1,FUNCT_2:def 3;
A3:dom T = the carrier of X by FUNCT_2:def 1;
    T" is LinearOperator of Y,X
    proof
      reconsider T1=T" as Function of Y,X by A1,A2,FUNCT_2:25;
A4: T1 is additive
      proof
        let y1,y2 be Point of Y;
        consider x1,x2 be Point of X such that
  A5: T1.y1=x1 & T1.y2=x2;
  A6: T.x1=y1 & T.x2=y2 by A5,A1,A2,FUNCT_1:32;
        x1+x2 =T1.(T.(x1+x2)) by A1,FUNCT_1:32, A3
             .=T1.(y1+y2) by A6, VECTSP_1:def 20;
        hence thesis by A5;
      end;
      T1 is homogeneous
      proof
        let y1 be Point of Y,r be Real;
        set x1 = T1.y1;
        r*x1=T1.(T.(r*x1)) by A1,FUNCT_1:32,A3
           .=T1.(r*T.x1) by LOPBAN_1:def 5
           .=T1.(r*y1) by A1,A2,FUNCT_1:32;
        hence thesis;
      end;
      hence thesis by A4;
    end;
    hence thesis by A1,FUNCT_1:33,A3;
  end;
