
theorem Th26:
  for X,Y be finite-dimensional RealLinearSpace
  holds
    [:X,Y:] is finite-dimensional
  & dim [:X,Y:] = dim X + dim Y
  proof
    let X,Y be finite-dimensional RealLinearSpace;

    consider BX be finite Subset of X such that
    A1: BX is Basis of X by RLVECT_5:def 1;
    A2: card BX = dim X by A1,RLVECT_5:def 2;

    consider BY be finite Subset of Y such that
    A3: BY is Basis of Y by RLVECT_5:def 1;

    reconsider BBX = [:BX,{0.Y}:]
      as Subset of the carrier of [:X,Y:] by ZFMISC_1:96;
    reconsider BBY = [:{0.X},BY:]
      as Subset of the carrier of [:X,Y:] by ZFMISC_1:96;

    reconsider BBX as finite Subset of the carrier of [:X,Y:];
    reconsider BBY as finite Subset of the carrier of [:X,Y:];

    reconsider BB=BBX \/ BBY as finite
    Subset of the carrier of [:X,Y:];

    BBX /\ BBY = {}
    proof
      assume BBX /\ BBY <> {};
      then consider x be object such that
      A4: x in BBX /\ BBY by XBOOLE_0:def 1;
      A5: x in BBX & x in BBY by A4,XBOOLE_0:def 4;

      consider a,b be object such that
      A6: a in BX & b in {0.Y} & x = [a,b] by A5,ZFMISC_1:def 2;

      consider a1,b1 be object such that
      A7: a1 in {0.X} & b1 in BY & x = [a1,b1] by A5,ZFMISC_1:def 2;

      a = a1 & b = b1 by A6,A7,XTUPLE_0:1;
      then
      0.X in BX by A6,A7,TARSKI:def 1;
      hence contradiction by Lm1,A1;
    end;

    then
    A9: card (BBX \/ BBY)
     = card BBX + card BBY by CARD_2:40,XBOOLE_0:def 7
    .= card BX + card BBY by CARD_1:69
    .= card BX + card [:BY,{0.X}:] by CARD_2:4
    .= card BX + card BY by CARD_1:69;
    A10: BB is Basis of [:X,Y:] by A1,A3,Th25;
    hence
    A11: [:X,Y:] is finite-dimensional by RLVECT_5:def 1;

    card BB
     = card BX + card BY by A9
    .= dim X + dim Y by A2,RLVECT_5:def 2,A3;
    hence thesis by A10,A11,RLVECT_5:def 2;
  end;
