
theorem Th30:
  for X,Y be RealLinearSpace-Sequence
  holds carr(X^Y) = (carr X) ^ (carr Y)
  proof
    let X,Y be RealLinearSpace-Sequence;
    reconsider CX = carr X, CY = carr Y as FinSequence;
    A1: len CX = len X
      & len CY = len Y
      & len carr(X^Y) = len(X^Y) by PRVECT_1:def 11;

    len carr(X^Y) = len X + len Y
    & len(CX^CY) = len X + len Y by A1,FINSEQ_1:22;

    then
    A2: dom carr(X^Y) = dom(CX^CY) by FINSEQ_3:29;

    for i be Nat st i in dom carr(X^Y)
    holds carr (X^Y).i = (CX^CY).i
    proof
      let i be Nat;
      assume
      A3: i in dom carr(X^Y); then
      reconsider i1 = i as Element of dom(X^Y) by A1,FINSEQ_3:29;
      A4: carr (X^Y).i = the carrier of (X^Y).i1 by PRVECT_1:def 11;
      A5: i in dom(X^Y) by A1,A3,FINSEQ_3:29;

      per cases by A5,FINSEQ_1:25;
      suppose
        A6: i in dom X; then
        A7: i in dom CX by A1,FINSEQ_3:29;
        reconsider i2 = i1 as Element of dom X by A6;
        thus carr (X^Y).i = the carrier of X.i2 by A4,FINSEQ_1:def 7
        .= CX.i by PRVECT_1:def 11
        .= (CX^CY).i by A7,FINSEQ_1:def 7;
      end;
      suppose
        ex n be Nat st n in dom Y & i = len X + n;
        then consider n be Nat such that
        A8: n in dom Y & i = len X + n;
        A9: n in dom CY by A1,A8,FINSEQ_3:29;
        reconsider n1 = n as Element of dom Y by A8;
        thus carr (X^Y).i = the carrier of Y.n1 by A4,A8,FINSEQ_1:def 7
        .= CY.n by PRVECT_1:def 11
        .= (CX^CY).i by A1,A8,A9,FINSEQ_1:def 7;
      end;
    end;
    hence carr(X^Y) = (carr X) ^ (carr Y) by A2,FINSEQ_1:13;
  end;
