reserve G,F for RealLinearSpace;

theorem Th11:
  for X be RealLinearSpace holds
  ex I be Function of X, product <*X*>
  st I is one-to-one & I is onto
  & ( for x be Point of X holds I.x = <*x*> )
  & ( for v,w be Point of X holds I.(v+w) = I.v + I.w )
  & ( for v be Point of X, r be Element of REAL
  holds I.(r*v)=r*(I.v) )
  & I.(0.X)=0.product <*X*>
  proof
    let X be RealLinearSpace;
    set CarrX = the carrier of X;
    consider I be Function of CarrX,product <*CarrX*> such that
    A1: I is one-to-one & I is onto
    & for x be object st x in CarrX holds I.x = <*x*> by Th4;
    len carr <*X*> = len <*X*> by PRVECT_1:def 11; then
    A2:len carr <*X*> = 1 by FINSEQ_1:40;
    A3:dom <*X*> = {1} by FINSEQ_1:2,def 8;
    A4:<*X*>.1 = X;
    1 in {1} by TARSKI:def 1; then
    (carr <*X*>).1= the carrier of X by A3,A4,PRVECT_1:def 11; then
    A5:carr <*X*> = <* CarrX *> by A2,FINSEQ_1:40; then
    reconsider I as Function of X,product <*X*>;
    A6:for x be Point of X holds I.x = <*x*> by A1;
    A7:for v,w be Point of X holds I.(v+w) = I.v + I.w
    proof
      let v,w be Point of X;
      A8:I.v = <*v*> & I.w = <*w*> & I.(v+w) = <*v+w*> by A1;
      reconsider Iv = I.v, Iw = I.w as Element of product carr <*X*>;
      1 in {1} by TARSKI:def 1; then
      reconsider j1=1 as Element of dom carr <*X*> by A2,FINSEQ_1:2,def 3;
      A10: (addop <*X*>).j1 = the addF of (<*X*>.j1) by PRVECT_1:def 12;
      A11: ([:addop <*X*>:].(Iv,Iw)).j1
      = ((addop <*X*>).j1).(Iv.j1,Iw.j1) by PRVECT_1:def 8
      .= v+w by A10,A8;
      consider Ivw be Function such that
      A12: I.v + I.w = Ivw & dom Ivw = dom carr <*X*>
      & for i be object st i in dom carr <*X*> holds Ivw.i in carr (<*X*>).i
      by CARD_3:def 5;
      A13: dom Ivw = Seg 1 by A2,A12,FINSEQ_1:def 3; then
      reconsider Ivw as FinSequence by FINSEQ_1:def 2;
      len Ivw = 1 by A13,FINSEQ_1:def 3;
      hence thesis by A8,A12,A11,FINSEQ_1:40;
    end;
    A14:for v be Point of X, r be Element of REAL holds I.(r*v)=r*(I.v)
    proof
      let v be Point of X, r be Element of REAL;
      A15:I.v = <*v*> & I.(r*v) = <* r*v *> by A1;
      1 in {1} by TARSKI:def 1; then
      reconsider j1=1 as Element of dom carr <*X*> by A2,FINSEQ_1:2,def 3;
      A17: (multop <*X*>).j1 = the Mult of (<*X*>.j1) by PRVECT_2:def 8;
      reconsider Iv = I.v as Element of product carr <*X*>;
      A18: ([:multop <*X*>:].(r,Iv)).j1
      = ((multop <*X*>).j1).(r,Iv.j1) by PRVECT_2:def 2
      .= r*v by A17,A15;
      consider Ivw be Function such that
      A19: r*(I.v) = Ivw & dom Ivw = dom carr <*X*>
      & for i be object st i in dom carr <*X*> holds Ivw.i in carr (<*X*>).i
      by CARD_3:def 5;
      A20: dom Ivw = Seg 1 by A2,A19,FINSEQ_1:def 3; then
      reconsider Ivw as FinSequence by FINSEQ_1:def 2;
      len Ivw = 1 by A20,FINSEQ_1:def 3;
      hence thesis by A15,A19,A18,FINSEQ_1:40;
    end;
    I.(0.X) = I.(0.X + 0.X)
    .= I.(0.X) + I.(0.X) by A7; then
    I.(0.X) - I.(0.X)
    = I.(0.X) + (I.(0.X) - I.(0.X)) by RLVECT_1:28
    .= I.(0.X) + 0.product <*X*> by RLVECT_1:15
    .= I.(0.X); then
    0.product <*X*> = I.(0.X) by RLVECT_1:15;
    hence thesis by A1,A6,A5,A7,A14;
  end;
