
theorem Th25:
  for X,Y be RealLinearSpace,
       BX be Basis of X,
       BY be Basis of Y
  holds
    [:BX,{0.Y}:] \/ [:{0.X},BY:] is Basis of [:X,Y:]
  proof
    let X,Y be RealLinearSpace;
    let BX be Basis of X,
        BY be Basis of Y;
    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 BB = BBX \/ BBY
      as Subset of the carrier of [:X,Y:];

    consider T1 be LinearOperator of X,[:X,(0).Y:] such that
    A1: T1 is bijective and
    A2: for x be Element of X
        holds T1.x = [x,0.Y] by Th23;

    for y be object
    holds y in T1.:BX iff y in BBX
    proof
      let y be object;
      hereby
        assume y in T1.:BX;
        then consider x be object such that
        A3: x in the carrier of X
          & x in BX & y = T1.x by FUNCT_2:64;
        A4: y = [x,0.Y] by A2,A3;
        0.Y in {0.Y} by TARSKI:def 1;
        hence y in BBX by A3,A4,ZFMISC_1:87;
      end;
      assume y in BBX;
      then consider a, b be object such that
      A5: a in BX & b in {0.Y} & y = [a,b] by ZFMISC_1:def 2;
      A6: b = 0.Y by A5,TARSKI:def 1;
      T1.a = [a,0.Y] by A2,A5;
      hence y in T1.:BX by A5,A6,FUNCT_2:35;
    end;
    then
    A7: T1.:BX = BBX by TARSKI:2;

    consider T2 be LinearOperator of Y,[:(0).X,Y:] such that
    A8: T2 is bijective and
    A9: for y be Element of Y
        holds T2.y = [0.X,y] by Th24;

    for y be object
    holds y in T2.:BY iff y in BBY
    proof
      let y be object;
      hereby
        assume y in T2.:BY;
        then consider x be object such that
        A10: x in the carrier of Y
           & x in BY & y = T2.x by FUNCT_2:64;
        A11: y = [0.X,x] by A9,A10;
        0.X in {0.X} by TARSKI:def 1;
        hence y in BBY by A10,A11,ZFMISC_1:87;
      end;
      assume y in BBY;
      then consider a, b be object such that
      A12: a in {0.X} & b in BY & y = [a,b] by ZFMISC_1:def 2;

      A13: a = 0.X by A12,TARSKI:def 1;
      T2.b = [0.X,b] by A9,A12;
      hence y in T2.:BY by A12,A13,FUNCT_2:35;
    end;
    then
    A14: T2.:BY = BBY by TARSKI:2;

    X is Subspace of X by RLSUB_1:25;
    then reconsider W1 = [:X,(0).Y:] as Subspace of [:X,Y:] by Th21;

    Y is Subspace of Y by RLSUB_1:25;
    then reconsider W2 = [:(0).X,Y:] as Subspace of [:X,Y:] by Th21;

    A15: BBX is Basis of W1 by A1,A7,REAL_NS2:87;
    A16: BBY is Basis of W2 by A8,A14,REAL_NS2:87;

      W1 + W2 = [:X,Y:]
    & W1 /\ W2 = (0). [:X,Y:] by Th22;

    hence [:BX,{0.Y}:] \/ [:{0.X},BY:] is Basis of [:X,Y:] by A15,A16,Th20;
  end;
