theorem Th4:
  (A + B) + C = A + (B + C)
proof
A1: Indices A= Indices ((A + B) +C) by MATRIX_0:26;
A2: Indices A= Indices (A + B) by MATRIX_0:26;
A3: Indices A= Indices B by MATRIX_0:26;
  now
    let i,j;
    assume
A4: [i,j] in Indices ((A + B) + C);
    hence ((A + B)+C)*(i,j)=(A+B)*(i,j) + C*(i,j) by A1,A2,Def5
      .=(A*(i,j) + B*(i,j)) + C*(i,j) by A1,A4,Def5
      .=A*(i,j) + (B*(i,j) + C*(i,j)) by RLVECT_1:def 3
      .=A*(i,j) + ( B + C)*(i,j) by A3,A1,A4,Def5
      .=(A + ( B + C))*(i,j) by A1,A4,Def5;
  end;
  hence thesis by MATRIX_0:27;
end;
