reserve G,F for RealLinearSpace;

theorem Th13:
  for X,Y be non empty RealLinearSpace-Sequence
  holds ex I be Function of [:product X,product Y:],product (X^Y)
  st I is one-to-one & I is onto
  & ( for x be Point of product X, y be Point of product Y
  holds ex x1,y1 be FinSequence st x=x1 & y=y1 & I.(x,y) = x1^y1 )
  & ( for v,w be Point of [:product X,product Y:]
  holds I.(v+w) = I.v + I.w )
  & ( for v be Point of [:product X,product Y:], r be Element of REAL
  holds I.(r*v)=r*(I.v) )
  & I.(0.[:product X,product Y:]) = 0.product (X^Y)
  proof
    let X,Y be non empty RealLinearSpace-Sequence;
    reconsider CX = carr X, CY = carr Y as non-empty non empty FinSequence;
    A1:len CX = len X & len CY = len Y
    & len carr (X^Y) = len (X^Y) by PRVECT_1:def 11;
    consider I be Function of [:product CX,product CY:],product (CX^CY)
    such that
    A2: I is one-to-one & I is onto
    & for x,y be FinSequence st x in product CX & y in product CY
    holds I.(x,y) = x^y by Th6;
    set PX = product X;
    set PY = product Y;
    len carr (X^Y) = len X + len Y & len (CX^CY) =len X + len Y
    by A1,FINSEQ_1:22; then
    A3:dom carr (X^Y) = dom (CX^CY) by FINSEQ_3:29;
    A4:for k be Nat st k in dom carr (X^Y) holds carr (X^Y).k = (CX^CY).k
    proof
      let k be Nat;
      assume A5: k in dom carr (X^Y); then
      reconsider k1=k as Element of dom (X^Y) by A1,FINSEQ_3:29;
      A6: carr (X^Y).k = the carrier of ((X^Y).k1) by PRVECT_1:def 11;
      A7: k in dom (X^Y) by A1,A5,FINSEQ_3:29;
      per cases by A7,FINSEQ_1:25;
      suppose A8: k in dom X; then
        A9: k in dom CX by A1,FINSEQ_3:29;
        reconsider k2=k1 as Element of dom X by A8;
        thus carr (X^Y).k = the carrier of (X.k2) by A6,FINSEQ_1:def 7
        .= CX.k by PRVECT_1:def 11
        .= (CX^CY).k by A9,FINSEQ_1:def 7;
      end;
      suppose ex n be Nat st n in dom Y & k=len X + n; then
        consider n be Nat such that
        A10:  n in dom Y & k=len X + n;
        A11: n in dom CY by A1,A10,FINSEQ_3:29;
        reconsider n1=n as Element of dom Y by A10;
        thus carr (X^Y).k = the carrier of (Y.n1) by A6,A10,FINSEQ_1:def 7
        .= CY.n by PRVECT_1:def 11
        .= (CX^CY).k by A11,A10,A1,FINSEQ_1:def 7;
      end;
    end; then
    A12:carr (X^Y) = CX^CY by A3,FINSEQ_1:13;
    reconsider I as Function of [:PX,PY:] ,product (X^Y) by A3,A4,FINSEQ_1:13;
    A13:for x be Point of product X, y be Point of product Y
    holds ex x1,y1 be FinSequence st x=x1 & y=y1 & I.(x,y) = x1^y1
    proof
      let x be Point of PX, y be Point of PY;
      reconsider x1=x, y1=y as FinSequence by Lm2;
      I.(x,y) = x1^y1 by A2;
      hence thesis;
    end;
    A14:for v,w be Point of [:PX,PY:] holds I.(v+w) = I.v + I.w
    proof
      let v,w be Point of [:PX,PY:];
      consider x1 be Point of PX, y1 be Point of PY such that
      A15: v = [x1,y1] by Lm1;
      consider x2 be Point of PX, y2 be Point of PY such that
      A16: w = [x2,y2] by Lm1;
      reconsider xx1=x1, xx2=x2 as FinSequence by Lm2;
      reconsider yy1=y1, yy2=y2 as FinSequence by Lm2;
      reconsider xx12=x1+x2, yy12=y1+y2 as FinSequence by Lm2;
      A17: dom xx1 = dom CX & dom xx2 = dom CX & dom xx12 = dom CX
      & dom yy1 = dom CY & dom yy2 = dom CY & dom yy12 = dom CY by CARD_3:9;
      I.v = I.(x1,y1) & I.w = I.(x2,y2) by A15,A16; then
      A18:I.v = xx1^yy1 & I.w = xx2^yy2 by A2;
      I.(v+w) = I.(x1+x2,y1+y2) by A15,A16,Def1; then
      A19:I.(v+w) = xx12^yy12 by A2; then
      A20: dom (xx12^yy12) = dom carr (X^Y) by CARD_3:9;
      reconsider Iv = I.v, Iw = I.w as Element of product carr (X^Y);
      reconsider Ivw = I.v + I.w as FinSequence by Lm2;
      A21:dom Ivw = dom carr (X^Y) by CARD_3:9;
      for j0 be Nat st j0 in dom Ivw holds Ivw.j0 = (xx12^yy12).j0
      proof
        let j0 be Nat;
        assume j0 in dom Ivw; then
        reconsider j=j0 as Element of dom carr (X^Y) by CARD_3:9;
        A22: Ivw.j0 = ((addop (X^Y)).j).(Iv.j,Iw.j) by PRVECT_1:def 8
        .= (the addF of (X^Y).j).(Iv.j,Iw.j) by PRVECT_1:def 12;
        per cases by A22,A3,FINSEQ_1:25;
        suppose A23: j0 in dom CX; then
          j0 in dom X by A1,FINSEQ_3:29; then
          A24:  (X^Y).j = X.j0 by FINSEQ_1:def 7;
          A25:  Iv.j = xx1.j & Iw.j = xx2.j by A23,A17,A18,FINSEQ_1:def 7;
          A26:  (xx12^yy12).j0 = xx12.j0 by A23,A17,FINSEQ_1:def 7;
          reconsider j1=j0 as Element of dom carr X by A23;
          (the addF of (X^Y).j).(Iv.j,Iw.j)
          =((addop X ).j1).(xx1.j1,xx2.j1) by A24,A25,PRVECT_1:def 12
          .= (xx12^yy12).j0 by A26,PRVECT_1:def 8;
          hence Ivw.j0 = (xx12^yy12).j0 by A22;
        end;
        suppose ex n be Nat st n in dom CY & j0=len CX + n; then
          consider n be Nat such that
          A27:    n in dom CY & j0=len CX + n;
          A28:   len CX= len xx1 & len CX = len xx2 & len CX= len xx12
          by A17,FINSEQ_3:29;
          n in dom Y by A1,A27,FINSEQ_3:29; then
          A29:  (X^Y).j = Y.n by A27,A1,FINSEQ_1:def 7;
          A30:  Iv.j = yy1.n & Iw.j = yy2.n by A17,A18,A27,A28,FINSEQ_1:def 7;
          A31:  (xx12^yy12).j0 = yy12.n by A27,A28,A17,FINSEQ_1:def 7;
          reconsider j1=n as Element of dom carr Y by A27;
          (the addF of (X^Y).j).(Iv.j,Iw.j)
          = ((addop Y).j1).(yy1.j1,yy2.j1) by A29,A30,PRVECT_1:def 12
          .= (xx12^yy12).j0 by A31,PRVECT_1:def 8;
          hence Ivw.j0 = (xx12^yy12).j0 by A22;
        end;
      end;
      hence thesis by A19,A20,A21,FINSEQ_1:13;
    end;
    A32:for v be Point of [:PX,PY:], r be Element of REAL holds I.(r*v)=r*(I.v)
    proof
      let v be Point of [:PX,PY:],r be Element of REAL;
      consider x1 be Point of PX, y1 be Point of PY such that
      A33: v = [x1,y1] by Lm1;
      reconsider xx1=x1, yy1=y1 as FinSequence by Lm2;
      reconsider rxx1=r*x1, ryy1=r*y1 as FinSequence by Lm2;
      A34: dom xx1 = dom CX & dom yy1 = dom CY
      & dom rxx1 = dom CX & dom ryy1 = dom CY by CARD_3:9;
      A35:I.v = I.(x1,y1) by A33 .= xx1^yy1 by A2;
      A36:I.(r*v) = I.(r*x1,r*y1) by A33,Def2 .= rxx1^ryy1 by A2;
      reconsider Iv = I.v as Element of product carr (X^Y);
      reconsider rIv=r*I.v as FinSequence by Lm2;
      A37:dom rIv = dom carr (X^Y) by CARD_3:9;
      A38: dom (rxx1^ryy1) = dom carr (X^Y) by A36,CARD_3:9;
      for j0 be Nat st j0 in dom rIv holds rIv.j0 = (rxx1^ryy1).j0
      proof
        let j0 be Nat;
        assume A39:j0 in dom rIv; then
        reconsider j=j0 as Element of dom carr (X^Y) by CARD_3:9;
        A40: rIv.j0 = ((multop (X^Y)).j).(r,Iv.j) by PRVECT_2:def 2
        .= (the Mult of (X^Y).j).(r,Iv.j) by PRVECT_2:def 8;
        per cases by A3,A39,A37,FINSEQ_1:25;
        suppose A41: j0 in dom CX; then
          j0 in dom X by A1,FINSEQ_3:29; then
          A42:  (X^Y).j = X.j0 by FINSEQ_1:def 7;
          A43:  Iv.j = xx1.j by A41,A34,A35,FINSEQ_1:def 7;
          A44:  (rxx1^ryy1).j0 = rxx1.j0 by A41,A34,FINSEQ_1:def 7;
          reconsider j1=j0 as Element of dom carr X by A41;
          (the Mult of (X^Y).j).(r,Iv.j)
          = ((multop X ).j1).(r,xx1.j1) by A42,A43,PRVECT_2:def 8
          .= (rxx1^ryy1).j0 by A44,PRVECT_2:def 2;
          hence rIv.j0 = (rxx1^ryy1).j0 by A40;
        end;
        suppose ex n be Nat st n in dom CY & j0=len CX + n; then
          consider n be Nat such that
          A45:    n in dom CY & j0=len CX + n;
          A46:   len CX= len xx1 & len CX= len rxx1 by A34,FINSEQ_3:29;
          n in dom Y by A45,A1,FINSEQ_3:29; then
          A47:  (X^Y).j = Y.n by A45,A1,FINSEQ_1:def 7;
          A48:  Iv.j = yy1.n by A35,A45,A34,A46,FINSEQ_1:def 7;
          A49:  (rxx1^ryy1).j0 = ryy1.n by A45,A46,A34,FINSEQ_1:def 7;
          reconsider j1=n as Element of dom carr Y by A45;
          (the Mult of (X^Y).j).(r,Iv.j)
          = ((multop Y ).j1).(r,yy1.j1) by A47,A48,PRVECT_2:def 8
          .= (rxx1^ryy1).j0 by A49,PRVECT_2:def 2;
          hence rIv.j0 = (rxx1^ryy1).j0 by A40;
        end;
      end;
      hence thesis by A36,A38,A37,FINSEQ_1:13;
    end;
    I.(0.[:PX,PY:]) = I.(0.[:PX,PY:] + 0.[:PX,PY:])
    .= I.(0.[:PX,PY:]) + I.(0.[:PX,PY:]) by A14; then
    I.(0.[:PX,PY:]) - I.(0.[:PX,PY:])
    = I.(0.[:PX,PY:]) + (I.(0.[:PX,PY:]) - I.(0.[:PX,PY:])) by RLVECT_1:28
    .= I.(0.[:PX,PY:]) + 0. product (X^Y) by RLVECT_1:15
    .= I.(0.[:PX,PY:]); then
    0. product (X^Y) = I.(0.[:PX,PY:]) by RLVECT_1:15;
    hence thesis by A13,A14,A32,A2,A12;
  end;
